Generalized measure theory

Foundations o f Physics, Vol. 3, No. 3, 1973

Generalized Measure Theory

Stanley Gudder

Depar tment of Mathematics, University of Denver Denver, Colorado

Received July 18, 1972

It is argued that a reformulation o f classical measure theory is necessary i f the theory is to accurately describe measurements o f physical phenomena. The postulates o f a generalized measure theory are given and the fundamentals o f this theory are developed, and the reader is introduced to some open questions and possible applications. Specifically, generalized measure spaces and integration theory are considered, the partial order structure is studied, and applications to hidden variables and the logic o f quantam mechanics are given.

1. INTRODUCTION

As important as modern measure theory is, there are indications that a reexamination of its basic postulates are in order. If measure theory is to describe accurately the processes taking place in finding lengths, areas, volumes, masses, charges, averages, etc. in practical situations and under conditions demanded by nature, its present structure must be generalized to include a wider class of phenomena than is currently applicable. In this introduction, we will attempt to illustrate this contention.

Suppose we are making length measurements with a micrometer and that this micrometer is accurate to within 10 .4 cm and can measure lengths up to 1 cm. Since we must round off measurements to the nearest 10 .4 cm, the micrometer in effect is able to give lengths only of the form v × 10 .4 cm, where n is an integer between 1 and 104. Let D be the interval [0, 1] and let C

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400 Stanley Gudder

be the class of subsets of X2 to which we can attribute definite lengths. Now, any interval of the form [m x 10 -4, n × 10-4], m ~< n, (m, n = 1, 2,..., 10 4) has the definite length (n - - m ) × 10 -4, so such intervals are in C. In the same way, any interval of the form

[ ~ + m x 10-4, c~ + n × 10-4],0 ~< c~ ~ 1 , ~ + n × 10 -4 ~< 1, m ~<n

= 1, 2,..., 10 4

is in C and has length (n - - rn) × 10 -4. We can also admit noninterval sets in C. For example, the set [0, 10 -4] u [2 x 10 -4, 3 × 10 -4] is in C and has definite length 2 × 10 -4. Although the length of the set

A = [ 0 , ½ x 10 -4 ] u [ 8 x 10 -4 , 2 x 10 -41

cannot be accurately measured directly, we can measure its complement [½ × 10 -4, ~- × 10 -4] w [2 × 10 -4, 1] (we assume the length of a point is zero) and conclude the length of A is 10 -4. Thus A ~ C. We thus conclude that C contains all intervals of the form [~ -t- m × 10 -4, ~ ÷ n × 10 -4] as above together with all sets that can be obtained f rom these using the operations of finite disjoint unions (ignoring points) and complementations. Notice that C is not closed under formations of unions or intersections of two arbitrary sets of C. The length of any member of C must have the form n × 10 -4, n = 1,..., 10 4.

Notice that if the length is to have decent properties, we are forced to assume that C has the above structure. For example, suppose we were to assume that any subinterval of [0, 1] is in C and that the length of a subinterval is that measured by the micrometer rounded off to the nearest 10 .4 cm. We would then conclude that the length of the interval [0, ½ × 10 -4] is zero, for example. Similarly, the lengths of [1 × 10-4, § × 10-4] and

[~ × 10 -4, 10-4]

are zero. I f the length is to have the eminently reasonable property of additivity, we would then be faced with the contradictory conclusion that the length of [0, 10 .4 ] is zero. As another example of what can go wrong in this case, suppose we assume that the length of [0, ½ × 10 -4] is unity. Using additivity, we would conclude that the length of (½ × 10 -4, 1] is zero and similarly the length of [0, ½ × 10 -4) is zero, so that the length of the point ½ × 10 -4 is unity!

One might argue that the situation we have considered is unrealistic in that we have unduly restricted ourselves. We do not have to use micrometers with an accuracy of only 10 .4 cm. We can, in principle, construct length- measuring apparatus with increasingly fine accuracy. We can then attribute

Generalized Measure Theory 401

to any interval an arbitrarily precise length, go through a similar construction as before (using countable unions instead of finite disjoint unions) and obtain the Lebesgue-measurable sets and Lebesgue measure on [0, 1]. Our answer to this argument is twofold. First, if we are to describe a particular measuring apparatus, we must be content with its inherent accuracy. Second, it is possible that nature has forced upon us an intrinsic limit to accurate length measurement. There is experimental as well as theoretical evidence (1.3,7,9) pointing toward the existence of an elementary length. This would be a length A (about 10 -15 cm) such that no smaller length measurement is attainable and all length measurements must be an integer multiple of 1. I f an "ult imate" apparatus could be constructed which can measure this length, then upon replacing 10 .4 cm by ;~, we would be forced to a construction similar to the above.

There are, in fact, instances in nature in which an ultimate accuracy is known to obtain. I t is accepted that the charge on the electron is the smallest charge obtainable. All charge measurements must give an integer multiple of e. I f we think, for example, of a charge of 104e as uniformly distributed over the interval [0, 104e], then a description of a charge measurement would begin with a construction similar to the above.

Another example of the above phenomenon is motivated by quantum mechanics. Suppose we are considering a particle which is constrained to move along the x axis. According to classical mechanics, if we form a two- dimensional phase space D with coordinate axes x, p, where p is the x-momentum of the particle, then the mechanics of the particle are completely described by a point in D. I f we want to describe quantum mechanical effects, however, we would have to contend with the Heisenberg uncertainty principle. (9,1°) For this reason, a point in ~ is physically meaningless since one can never determine whether the phase space coordinates (x, p) of a particle are at a particular point. The best one can do is determine if (x, p) is contained in a rectangle with sides of length ,dx, Ap where Ax Ap = h/2. Let us suppose our measuring instruments have ultimate precision so we can determine if (x, p) is in a rectangle of area h/2. Physically, these elementary rectangles in f2 become the basic elements of the theory instead of the points in f2. Generalizing this slightly, we say that a set E is admissible if the area of E (strictly speaking, we mean the Lebesgue measure of E ) i s an integer multiple of h/2. An admissible set E is one for which it is physically meaning- ful to say that (x, p ) s E. For simplicity, let us assume that D is a large rectangle of area nh/2, where n is a large integer. Let us now consider the mathematical structure of the class C of admissible sets. First, it is clear that f2 E C and also ~ ~ C. I t is clear that if E ~ C, then the complement E' ~ C. Further if E, F e C are disjoint, then E w F ~ C. Note, howe~er, that in general if E, F ~ C, then E c~ F and E u F need not be in C.

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402 Stanley Gudder

We hope it is now clear that to study certain diverse practical situations, the basic f r amework of measure theory should be generalized. Ins tead of considering measure spaces (£2, 6g,/x) in which ~ is a a-a lgebra of subsets of a set £2 and/x is a measure on 6g, it m a y be useful to study structures (£2, C,/~) where C is a more general class of subsets o f £2 than a a-algebra.

In this article, we develop some of the fundamenta ls of this generalized measure theory and introduce the reader to some open questions and possible applications.

2. GENERALIZED MEASURE SPACES

Let £2 be a nonempty set. A a-class C of subsets of £2 is a collection of subsets which satisfy (i) £2 e C; (ii) if a e C, then a ' e C; (iii) if ai ~ C are mutual ly disjoint, then Uai e C, i = 1, 2,...; where we denote the comple- men t of a set a by a ' . A measure t ~ on C is a nonnegat ive set funct ion on C such t h a t / ~ ( ~ ) = 0 and/~(Uai) = ~ /~(a~) if as are mutual ly disjoint elements of C. A generalized measure space is a triple (£2, C,/z) where C is a a-class of subsets of £2 and/~ is a measure on C. The germ of this idea m a y be traced to P. Suppes. a3~ Fur ther studies are carried out in Refs. 8 and 12.

We say tha t two sets a, b ~ C are compatible (written a +-+ b) if a n b e C. A collection of sets A in C is said to be compatible provided any finite inter- section of sets in A belongs to C. Compat ib i l i ty is an impor tan t concept in this theory. Compat ib le sets correspond to noninterfering events in q u a n t u m mechanics.

Recall tha t a measure space is a triple (£2, 5 , tz) where 6~ is a cr-algebra of subsets o f £2, tha t is, (i) £2 ~ 6~; (ii) if a e 6~, then a ' e 6~; (iii) if a~ is a sequence in 6~, then Uai ~ 6~; a n d / z is a measure on 6~. I t is shown in Ref. 8 tha t a a-class is a a-a lgebra if and only if all its elements are pairwise compat ible . We now give some examples of a-classes tha t are not a-algebras. By placing a measure on these (which can always be done in a nontr ivial way), we obtain generalized measure spaces which are not measure spaces.

Example 1. The simplest a-class that is not a a-a lgebra is given as follows. Let £2 = {1, 2, 3, 4} and let C be the class of subsets of £2 with an even number of elements. Then C is a a-class but {1, 2} n {2, 3} = {2} ¢ C, so C is not a a-algebra. I t is easy to see tha t any pairwise compat ib le collection o f sets in C is compatible .

Example 2. Let £2 = {1, 2,..., 8} and let C be the class of subsets of £2 with an even number of elements. Again C is a a-class but is not a a-algebra. In this example, there are pairwise compat ib le collections which are not compat ible; for instance, {1, 2, 3, 4}, {1, 2, 5, 6}, {1, 3, 6, 8} is such a collection.


Example 3. A generalization of Examples 1 and 2 is the collection of Lebesgue-measurable subsets of the real line R with even measure. More generally, let )t > 0 and let D = [0, nA], where n is a positive integer. If C is the collection of Lebesgue-measurable subsets of £2 whose Lebesgue measures are an integer multiple of A, then C is a a-class but not a a-algebra.

Example 4. Let (Da, ~1) and (£22 , 692) be (generalized) measurable spaces [i.e., 6g~ and ~ are (a-classes) a-algebras in £21 and D2, respectively]. Let D~ = D1 × ~22 and let C = {Ax X £22, ~Q~ × A2 : A1 z 6~x, A2 z ~ } . Then C is a a-class which is not a a-algebra if (~a, ~ ) and (~Q2,6~) are nontrivial. We call (~3, C) the a-class product of (.Q~, ~Y~) and (~Q2, ~2). The construction can be carried through for any finite number of measurable spaces or generalized measurable spaces.

Theorem 2.1, whose proof we omit, is an improvement upon Theorem 3.1 of Ref. 8.

Theorem 2.1. If {As : a e/1} is any compatible collection of sets belonging to a a-class C, then this collection is contained in a sub-a-algebra of C.

Let/z be a measure on a a-class C. Let us briefly compare some of the properties o f /z to those of a measure on a a-algebra. Now, a a-algebra measure v always satisfies v(A u B) + v(A n B) = v(A) + v(B). Our measure tz also satisfies this condition when the left-hand side is defined. Indeed, we would then have A n B e C, so by Theorem 2.1, A and B are contained in a sub-a-algebra 6g of C and the restriction o f /z to 6g is an ordinary measure. Now, a measure v on a a-algebra ~ is always subadditive; that is, v(U~ ° Ai) <~ Z [ v(Ai), for any A i e 6g. We now show that/z does not necessarily have this property when the left-hand side is defined. Consider the following subsets of the real line:

A = [0, 4], B = [2, 5], D = [0, 1] w [2, 3] u [5, 6].

Let C = {~, [0, 6], A, B, D, A', B', D'}. Then C is a a-class of subsets of [0, 6]. Define the measure/z on C as follows:

/z(¢) = 0,/z(A) =/z(B) =/z(C) = 1,/x(A') =/~(B') =/z(C') = 3,/x([0, 6)] = 4.

Now,

A u B k3 C = [0, 6] e C and yet ~([0, 6]) = 4 > 3 =/x(A) -k /x(B) +/~(C).

A measure /z on a a-class C is always subadditive on two sets; that is, /x(A u B) ~</~(A) -k Iz(B) for all A, B c C for which A t3 B ~ C. Indeed, if

404 Stanley Gudder

A u B ~ C, then A' n B' = (A u B) '~ C so A ' ~ B'. It follows [Lemma 2.30) of Ref. 8] that A ~ B and so by Theorem 2.1, A, B are contained in a sub-a-algebra of C.

I t follows f rom the nonsubadditivity of the previous paragraph that a measure/x on a a-class C cannot in general be extended to the a-algebra generated by C.

3. INTEGRATION THEORY

In this section, (g?, C,/x) will be a generalized measure space, R the real line, and B(R) the Borel a-algebra on R. A f u n c t i o n f : D - + R is measurable i f f - l ( E ) ~ C for every open set E _C R. I t is shown in Ref. 12 that the a-class generated by the open sets of any topological space is a a-algebra. I t follows that the a-class generated by the open sets of R is B(R). One can then conclude that if f is measurable, f - l ( E ) ~ C for every E ~ B(R). I f f is measurable, we use the notation Af = {f-~(E) : E ~ B(R)}. It is easily seen that A I is a sub-a-algebra. We say that two measurable functions f and g are compatible {written f+-+ g) if Af u Ag is compatible. I t follows f rom Theorem 2.1 that f+-+ g if and only i f f -~ (E) +-~ g-~(F) for every E, F ~ B(R). It is well known that in measure spaces, the sum of any two measurable functions is measurable. I t follows that in a generalized measure space, the sum of two compatible measurable functions is measurable. However, the sum of two noncompatible measurable functions need not be measurable. This follows from the general fact that C is a a-algebra if and only if the sum of any two measurable functions is measurable. (8) One c a n show that the sum of two noncompatible measurable characteristic functions is never measurable. ~S) However, there are noncompatible measurable functions whose sum is measurable. For instance, two such functions would be

f (1 ) = f (2 ) = f (7 ) = f (8 ) = 0, f (3) = f (4) = 1, f (5) = f (6 ) = 2;

g(1) = g(6) = 1, g(2) = g(4) = 2, g(3) = g(5) = g(7) = g(8) = 0

in Example 2. I f f is measurable, then/x restricted to A± is a measure on a a-algebra.

Thus (f2, AI, /x) becomes a measure space and we define the integral ffdl~ in the usual way. Let us discuss the basic properties of this integral. It is clear that ~[fd/z ~> 0 i f / > ~ 0 and f ~fd/z = c~ f fdtz for any ~ ~ R. Also iff+-~ g, then f ( f + g) d/z = ffdk~ + f g dtz. Of course, in ordinary integration theory, on a a-algebra, the integral is always additive (unless f fdlx = oo, f g d/z = - - 0% or vice versa). We now inquire whether this property holds for our generalized integral.


We say tha t two measurable functions f and g are summable i f f -t- g is measurable . We say that the integral is additive on summable functions

f a n d g if f ( f + g) d/~ = f f dt z + f g dtz (unless f f dtz = 0% f g dlx = -- oo, or vice versa). Now, it is clear that i f g is a constant a n d f i s measurable, then the integral is additive on f and g since f+-~ g. Al though one can show tha t the integral is additive on two summable functions if one of the functions has one or two values, it is not known, in general, whether the integral is additive on summable functions. However , we will later give a sufficient condit ion for this result to hold. Al though it would be an impor tan t mathemat ica l result if it were true that the integral is additive on summable functions, f rom a physical point of view, it would not be so surprising if this result did not hold in general. Indeed, measurable functions correspond to observable operations. Now, if two observable operat ions f and g are interfering, then even though the s u m f + g m a y be observable, it will have no relat ionship to the original observations. Thus there is no physical reason to expect the average of

f + g to be the average of f plus the average of g. Let f and g be summable simple functions with values ~1 ..... a,, and

f~l , ' " , / 3 m , respectively. I f c~ +/3~ = c% +/3~ for i v~ k, j va l, then we call (c~, fi~ ; c~7~ ,/3~) a degeneracy for f , g. I f the values o f f , g can be reordered in such a way that whenever ( ~ , flj ; c~e,/3,) is a degeneracy we have i = l, j = k, then f , g are called symmetric.

Lemma 3.1. I f f , g have no degeneracies, then the integral is additive on them.

Proof. f-l{c~i} n g-l{/3j} = ( f + g ) - i {(~i "~ /3j} ~ C; i = 1 ..... n; j = 1 ..... m; s o f ~ - + g.

Although, as we have shown in L e m m a 3.1, nondegenerate summable simple functions are compat ible , symmetr ic functions need not be compatible. Fo r example, let D = {1, 2,..., 9}, F1 = {1, 4, 7}, F2 = {2, 5, 8}, F 3 = { 3 , 6 , 9 } , G 1 = { 1 , 2 , 3 } , G~ = { 4 , 5 , 6 } , G~ = { 7 , 8 , 9 } , H 1 = { 2 , 4 } , //2 = {3, 7}, and H~ = {6, 8}, and let C be the a-class generated by these subsets together with the subsets {1}, {5}, {9}. Define the functions f , g by f (1 ) = f ( 4 ) = f ( 7 ) = 0, f ( 2 ) = f ( 5 ) ---- f ( 8 ) = 1, f ( 3 ) = f ( 6 ) = f ( 9 ) = 2, g(1) = g(2) = g(3) = --1, g(4) = g(5) = g(6) = O, g(7) = g(8) = g(9) = 1. I f we let (X 1 = 0 , O~ 2 = 1, % = 2,/31 = --1,/32 = 0, t38 = 1, then

+ P2 = + / 3 1 , = = + 8 2 .

I t is easily seen that f , g are measurable and summable , so l , g are symmetric. However , f ~ - ~ g since f - 1 {0} c~ g-1 {0} ~ C, for example. I t would be interesting to test the additivity of the integral in this case, so let us place a measure on C. Define the set function /z 0 by /z0{1 } =/z0{5 } =/z0{9 } = 0,

406 Stanley Gudder

/~o(F1) = 2, /~o(F~) = 0, /zo(F3) = 5, /zo(Ga) =/zo(G~) = 3, /~o(G3) = 1, /Lo(H1) = 1,/xo(H2) = 4, tzo(H~) = 2 and extend/~o to /x on C. We then see that . [ ( f + g ) d l ~ = 8 = .[ fdtz-k fgdtz , so we have additivity. Our next theorem shows that this always happens.

Theorem 3.1. I f f , g are symmetric, then f ( f + g) dl x = f f dl~ + f g dt~.

We omit the somewhat tedious p roof o f this theorem. Let us note that a l though the integral may not be additive, in general, on

summable functions there are interesting, special situations in which it is. For instance, in Example 4, Section 2, it can be shown that two simple functions are summable if and only if they are compatible. As another example, let/~ be a point measure on a or-class C; that is, there is a point

e D such that/x(A) = 1 if ~ 6 A,/x(A) = 0 if ~ ¢ A for all A e C. Then the integral with respect t o / x is always additive for summable functions. This same conclusion also holds if/x is a convex combinat ion (finite or infinite) o f point measures.

There is a simple basic proper ty that breaks down as far as the generalized integral is concerned. Suppose f and g are measurable functions in the generalized measure space (D, C,/x) which are equal almost everywhere; that is, {co : f(co) =/= g(oJ)} ~ C and /z{co : f(co) @ g(co)} = 0. Does f f d l x = f g dt~ hold? The following example shows that the answer can be n o .

Let f2 = {1, 2, 3, 4}, A1 = {1, 2}, Az = {3, 4}, B1 = {1, 3}, B2 = {2, 4}, //1 = {1, 4}, / /2 = {2, 3}. Then these sets and ~ fo rm a C-class in f2. Define a measure /~ on C by/x(A1) =/~(B2) = 1, /x(A2) =/x(B1) = 2, /x(H~) = 3, /x(H2) = 0. Then the characteristic functions XA~, X~ are equal almost everywhere but J" x ~ d/~ = 2 :/= 1 = f X,~ d/x.

4. PARTIAL ORDER STRUCTURE OF o-CLASSES

I f C = {a, b, c,...} is a a-class, then C is a partially ordered set (poset) under the natural order a ~ b if a __C_ b. Also, C has first and last elements 4, f) which we denote by 0, 1, respectively. As in any poset, we define the greatest lower bound (denoted a ^ b) and the least upper bound (denoted a v b) o f a and b in the usual way. Now, a n b and a v b need not exist; however, if a n b E C or a to b ~ C, it is easily seen that a ^ b = a n b and a v b = a u b. I f a ~ b q~ C (or a u b ql C), we still may have a ^ b (or a v b) existing. For instance, in Example 1, Section 2, {1, 2} n {2, 3} = {2} ¢ C but {1, 2} ^ {2, 3} = s~. We say that a and b are or thogonal (denoted a_ l_b) i f a <~b'. Notice i r a ± b , then b ± a , and a n b = a A b = ~ .


However , a ^ b = ;~ does not imply a 5_b. Indeed, in Example 1, {1, 2} ^ {2, 3} = ~ ye t{ l , 2} ~_ {2, 3}.

I f we let the pr ime denote the usual complement , then (C, ~<, ') becomes a a-orthocomplementedposet/2~ Tha t is, (1) a ~ b implies b' ~ a' , (2) a" = a, (3) a v a ' = 1, (4) if ai J_ a j , i v ~ j = 1, 2 ..... then Vai exists. Fur thermore , we have the following result.

Lemma 4.1. (C, ~<, ') is a a -o r thomodu la r poset, that is, if a ~< b, t h e n b = a v ( b ^ a ' ) .

Proof. I f a ~ b , then a n b ' = ~ , so a u b ' ~ C . Hence b n a ' = ( a u b ' ) ' ~ C , so b ^ a ' = b n a ' . Since a n ( b n a ' ) = ~ , we have a u (b n a ' ) ~ C , s o a v (b ^ a') = a u ( b n a ' ) = b .

N o w it seems tha t a a-class C has, in general, no further part ial order propert ies than its being a a -o r thomodu la r poset. Fo r example, C need not be a lattice (a poset in which a ^ b and a v b always exist). Indeed, in Example 2, { 1, 2, 3, 4} ^ {2, 3, 4, 5} does not exist. Even a-classes which are lattices need not be distributive and hence cannot be Boolean algebras/2) Indeed, Example 1 is a lattice, bu t{ l , 2} v ({1, 3} ^ {3, 4}) = ~ and

({1, 2} v {1, 3}) ^ ({1, 2} v {3, 4}) = {1, 2, 3, 4}.

Also, a-classes that are lattices need not have the weaker p roper ty of mod- ularity. 12) Fo r example, let 51 , 6g 2 be the a-algebras of all subsets o f {1, 2, 3} and {1, 2}, respectively, and let C be the a-class p roduc t 6g 1 × g/z. Let

a = {(1, 1), (1, 2)},

b = {(1, 1), (2, 1), (3, 1)},

c = {(1, 1), (1, 2), (2, 1), (2, 2)}.

T h e n a ~< c, b u t a v ( b A c ) = a :/= c = ( a v b ) ^ c . In the next section, we shall give an example of a a -o r thomodu la r poset

which is not i somorphic to a a-class. We now give an answer to the question of when a a -o r thomodu la r poset is ( i somorphic to) a a-class. A state on a a -o r thomodu la r poset ~a is a m a p m f rom ~ to the real unit interval [0, 1] tha t satisfies (1) m(1) = 1, (2) m(Va~) = 32 m(ai) i f a i ~ aj ' , i v a j = 1, 2 ..... A state is dispersion-free if its only values are 0 and 1. A set o f states M on is order determining if re(a) <~ re(b) for all m ~ M implies a ~< b.

Theorem 4.1. A a -o r thomodu la r poset ~ is i somorphic to a a-class if and only if the set S of dispersion-free states on ~ is order-determining. In this case, ~ is i somorphic to a a-class of subsets o f S.

408 Stanley Gudder

Proof. Suppose ~ is isomorphic to a a-class C of subsets of £2 under the isomorphism q~ : ~ ~ C. Suppose a, b e ~ and re(a) <~ re(b) for all m ~ S. In particular, if co ~ ¢(a) and /~o~ is the point measure at co, then c - -~ /~(¢(c) ) is a dispersion-free state, so /~(¢(a))~</x~(¢(b)) . Hence co e ¢(b) and ¢(a) ~< ¢(b). It follows that a ~ b, so S is order-determining. Conversely, suppose S is order-determining. Let h be the map f rom ~ to the collection of subsets o f S defined by h(a) = {me S: re(a) = 1} and let C be the range o f h. We first show C is a a-class. Now, S = h(1) and ~ = h(0) so S, ¢ e C. For h(a) e C, we have h(a)' -= h(a') e C. Also if h(ai) are mutual ly disjoint, we have L)~ h(ai )= h(Vai)e C. Now, we already have that h preserves '. I f a ~< b, then, clearly, h(a) <~ h(b). To show h is one-one, suppose a =/= b. Since S is order-determining, there is an m e S such that re(a) v L re(b). Thus re(a)= 1, re(b)= 0, or vice versa, so h(a) v a h(b). Finally, if h(a) <~ h(b), then re(a) <~ re(b) for all m ~ S and hence a ~< b.

I f S is not order-determining (in the next section, we give an example in which S = 2~; there are even examples of a -o r thomodula r lattices with no states(6)), then ~ cannot be isomorphic to a a-class. However, we now give a weaker result tha t is the analog of Loomis ' s theorem (n) which states that a Boolean a-algebra is a a -homomorph ic image of a a-algebra o f subsets.

Theorem 4.2. A a-or thocomplemented poset ~ is a one-one a -homomorph ic image of some e-class o f subsets.

Proof. Let ~ be the collection o f all maps f rom ~ into {0, 1} _ R such that co(a') = 1 - - co(a) for all a e ~ . Let ~(a) = {co e ~ : co(a) = 1}, a e ~ . The family C consisting o f £2, 4, and {¢(a): a ~ ~} is a e-class. Indeed, ¢(a) ' = ¢(a ' ) e C for all a e ~ . Now, ¢(a) n ¢(b) = ~ if and only if a = b'. Therefore, ¢(ai), i = 1, 2,..., are mutual ly disjoint if and only if i runs th rough the set {1, 2} with al = a2'. Thus U¢(ai) -= ¢(ai) t~ ¢(a2) = £2 e C. Now, define h : C ~ ~ by h(f2) = 1, h(~) = 0, h(¢(a)) = a, a e ~ . I t is easy to check that ¢ is a one -one a -homomorphism.

5. A P P L I C A T I O N S TO Q U A N T U M M E C H A N I C S

In general axiomatic quan tum mechanics, the set of proposi t ions for a quan tum system is taken to be a a -or thomodula r poset or lattice (1°,14) P = {a, b, c,...}. The quan tum states on P are states as defined in the previous section. The quan tum observables are defined to be a -homomorphisms f rom the real Borel sets B(R) to P. Two proposit ions a, b e P are said to be compatible if there are mutual ly disjoint elements a l , b~, c such that a = a~ v c, b = bl v e. Now, we see that a a-class C is a special case o f a quan tum proposit ional system and that quan tum states correspond to probabili ty


measures on C. Furthermore, it can be shown (8) that the two definitions of compatibility are equivalent on C. Finally it follows f rom a theorem due to Sikorsky and Varadarajan (14) that if x : B(R) -+ C is an observable, then there is a unique measurable function f : g?-+ R such that x(E) = f - l ( E ) for all E c B(R). Thus our generalized measure theory can be viewed as a particular instance of a quantum propositional system and all our measure- theoretic constructs, such as measures, compatibility, and measurable functions, correspond to important quantum mechanical notions. Now, the distinguishing feature of (r-classes among general (r-orthomodular posets is that (r-classes have an order-determining set of dispersion-free states. This might be interpreted as corresponding to the existence of "hidden variables" in the theory. (~a°) We feel that a generalized measure theory may give a useful model for quantum mechanics. For instance, a measurable function in Example 3 must have discrete values, so a "quantization" effect already appears.

Now, conventional quantum mechanics is also a special case of a quantum propositional system. In conventional quantum mechanics, a°.14) the quantum propositions are taken to be the (r-orthomodular lattice L(H) of closed subspaces (or equivalently, orthogonal projections) of a separable complex Hilbert space H. The order is defined as subspace inclusion and the complement is taken to be the orthogonal complement. An observable is now a projection-value measure x:B(R)-- , -L(H) which by the spectral theorem a°) corresponds to a unique self-adjoint operator A = J" hx(dh). It follows from a theorem due to Gleason (5) that corresponding to every state m on L(H) (if dim H > 2) there is a unique positive trace class operator T on H such that m(P) = tr(TP) for all P ~ L(H).

Probably the simplest quantum system is a photon. This system, in conventional quantum theory, is represented by a two-dimensional Hilbert space / /2 . I t is easily seen that L(H~) is isomorphic to a (r-class product I-I {C~:i ~ [0, 1]}, where Ci is the four-element Boolean algebra, i e [0, 1]. For dimension greater than two, however, L(H) is not isomorphic to a (r-class.

Theorem 5.1. I f dim H > 2, then L(H) is not isomorphic to a e-class.

Proof. I f L(H) were isomorphic to a (r-class, then by Theorem 4.1, there is a dispersion-free state rn on L(H). By Gleason's theorem, there is a trace class operator T such that re(P) = tr(PT) for all P ~ L(H). I f ;~i and ~i are the eigenvalues and orthonormal eigenvectors of iv, it follows that re(P) = Z Ai(~i, Pc~) for all P eL(H). I f P~ is the one-dimensional projection on q~s, we have

As = As I(q~J, ~,bs)l 2 = 52 A~(d?i, P~/?~) = m(P~) = 0 or 1.

410 Stanley Gudder

Since 1 : re(T) = ~ A i , we see tha t all the Ai are zero except one, so T is itself a one-d imens iona l p ro jec t ion P~. Now, if 5b is a uni t vector, we have l (¢ , ¢ ) i 2 : tr(PcP~) : m(P~) : 0 or 1. This is clearly impossible .

Let us briefly consider in tegra t ion theory in convent iona l qua n tum mechanics. I f A is a self-adjoint ope ra to r wi th spectral reso lu t ion pA(.) and m is a state on L(H), the integral of A relative to m is defined as I(A) : f )tra(pA(d)t)) when this quant i ty exists. Now, unlike the a-class case, two bounded self-adjoint opera to rs are always summable . This is because the sum of b o u n d e d self-adjoint opera to rs is self-adjoint. Now, there is no physical reason for this to be the case. This happens only as a result of the r ich Hi lbe r t space s t ructure and we feel this is an accident of the par t icu la r theory o f convent iona l q u a n t u m mechanics. W e now ask whether the integral I is addit ive; tha t is, does I(A ÷ B) : fAm(pA+B(dA)) : fAm(pA(dA)) + f )tm(PO(dA)) = I(A) + I(B) hold for any bound e d self-adjoint opera tors A, B (we consider only b o u n d e d opera tors , to avoid tedious p rob lems of domains and existence) ? Now, it is no t at all obvious whether this is the case. However , using the deep theo rem of Gleason cited above, we can show tha t this holds. Indeed, there is a t race class ope ra to r T such tha t m(P) : t r (TP) for all P ~ L(H), so I(A) : f A tr[TpA(dA)] : t r [T f ApA(dA)] : t r(TA). Hence I(A + B) : t r [T(A + B)] : t r (TA + TB) : t r TA + t r TB : I(A) + I(B). Aga in there is no intr insic physical reason for this to occur and we feel it is an accident o f the par t i cu la r theoret ica l model . I t m a y be tha t a more appro - pr ia te mode l for q u a n t u m mechanics might be fo rmula ted wi thin a general ized measure- theore t ic f ramework .

REFERENCES

1. D. Atkinson and M. Halpern, Non-usual topologies on space-time and high-energy scattering, Y. Math. Phys. 8, 373-387 (1967).

2. G. Birkhoff, Lattice Theory (A. M. S. Coll. Publ. XXV, Providence, R.I. 1967). 3. R. Blumenthal, D. Ehn, W. Faissler, P. Joseph, L. Langerotti, F. Pipkin, and D. Stairs,

Wide angle electron-pair production, Phys. Rev. 144, 1199-1223 (1966). 4. D. Bohm and J. Bub, A proposed solution of the measurement problem in quantum

mechanics by hidden variables, Rev. Mod. Phys. 38, 453-469 (1966). 5. A. Gleason, Measures on closed subspaces of a Hilbert space, J: Rat. Mech. Anal

6, 885-893 (1957). 6. R. Greechie, Orthomodular lattices admitting no states, Y. Comb. Theory 10, 119-132

(1971). 7. S. Gudder, Elementary length topologies in physics, SIAM J. Appl. Math. 16, 1011

1019 (1968). 8. S. Gudder, Quantum probability spaces, Proe. AMS 21, 296-302 (1969). 9. W. Heisenberg, The Physical Principles of Quantum Mechanics (Univ. of Chicago

Press, Chicago, II1., 1930).


10. J. M. Jauch, Foundations o f Quantum Mechanics (Addison-Wesley, Reading, Mass., 1968).

11. L. Loomis, On the representation of ~-complete Boolean algebras, Bull. A M S 53, 757-760 (1947).

12. T. Neubrunn, A note on quantum probability spaces, Proe. A M S 25, 672-675 (1970). 13. P. Suppes, The probabifistic argument for a nonclassical logic of quantum mechanics,

Philos. Sei. 33, 14-21 (1966). 14. V. Varadarajan, Geometry o f Quantum Theory, Vol. 1 (Van Nostrand, Princeton,

N. J., 1968).

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