An Introduction to Measure

A measure, m, on a domain D is a function (MD|m:S) from a collection, MD, of sub-domains of D to some additive domain for which: the union of all MD's members is D; MD is closed under finite union (iethe union of any finite sub-collection of MD is a member of MD); whenever the intersection, N, of two sets A and B, with union U, is also in MD, then m(A) + m(B) is equal to m(U) + m(N). We call members of MD `measurable' sub-domains of D; we refer to MD as m-Measurable(D). The conditions may thus be described as: any finite union of measurables is measurable; when the intersection of two measurable sets is measurable, the sum of its measure with that of their union is equal to the sum of their measures.

The additive domain need only support an addition on its members: the result of addition need not necessarily lie in the same domain (as, for instance: the sum of two projections is not, in general, a projection; a measure could yield a linear projection as its value and have sums of items be linear maps but rarely projections). To put it another way, with S truly an additive domain (so sums of its members lie in it), (MD|m|) may be some special restricted sub-domain of S.

Unions are here kept finite to allow D to be an m-infinite domain, in the sense that for any member of S there is some measurable sub-domain whose measure exceeds the given member of S (with respect to some suitable partial (or better) ordering on S). Then D need not itself be a member of MD, as long as MD is itself infinite. If we ask for all collections of measurable sub-domains to have measurable union, we can still describe bounded measures (below).

We say a measure is bounded on D if every sub-collection of MD has measurable union. In the event of S being an additive domain with infinities, one might be most interested in finite bounded measures: but this is not guaranteed, and can be thus distinguished, so I don't force the definition to have a finite bound, merely to have a bound in S.

It is usual to insist that MD include D's empty sub-domain with m({}) = 0. Given my aversion to requiring an explicit additive identity (eg in a positive scalar domain), I note that if empty were measurable, and any other A were measurable, then A's union with empty is A and intersection with empty is empty: whence A and empty have measurable intersection and the sum of m's values thereon and on thier union is trivially equal to the sum of m's values on A and empty themselves.

When S is a space of projectors for some symmetric antilinear iso-automorphism, we can take: measures, a and b, of some arbitrary pair of measurable sets with measurable intersection; the measure, n, of this intersection; and that, u, of the union of the given pair. Squaring both sides of a+b=n+u we obtain a+ab+ba+b = n+nu+un+u, whence (now cancelling a+b=n+u from each side), ab+ba = nu+un. The anticommutator of the measures of a pair of sets, like the sum of its measures, depends only on its intersection and union.

Transcript: Probability Measures

When our measure's yield-space include something `suitably like' the real interval [0,1], we describe the measure as a probability measure precisely if its values all lie in this `interval'. The required properties of this interval, I, are that: for every member of I, there is some s for which the member is s*s (the product of s with its conjugate); for every member of I, there is some other member of I whose sum with it is 1.

The projectors in a Hilbert space resemble such an interval, provided we can find, for each P, an antilinear square root, S, of 1 commuting with P. The product SP is then antilinear, so the appropriate notion for its product with its conjugate is, in fact, its square (since it conjugates that on which it acts, in this case itself): and SPSP = PSSP = PP =P since SS=1 and PP=P. The fact that 1-P is also a projector (and, in fact, it can use the same S to show its `positivity') completes the notion of the interval. It is perhaps necessary that S be obliged to have some suitable relationship with the antilinear g elided from `g-projector' whenever we described things as projectors.

Transcript: Integration

For: a measure (MD|m:V), with MD a collection of subsets of some domain D and V a vector space over some scalar domain S; a function (D|f:U), with U a vector space over S; and a bilinear multiplication (U×V|:P), with P yet another S-vector space: we say that f is integrable with respect to m precisely if there is a measure (MD|mf:P) for which every A in MD has mf(A) in m(A).convexHull(A|f|). Such a measure, mf, is said to integrate f with respect to m; and f is described as the density of mf with respect to m.

convexHull is defined as a function taking any subset of a vector space and yielding the minimal convex subset of that vector space containing the given subset. A subset, R, of a vector space is said to be convex precisely if, for any t between 0 and 1 and every r,s in R: tr+ (1-t)s is also in R. The intersection of any collection of convex sets is inevitably convex: the convex hull of a set is just the intersection of all convex sets containing that set. Discussion of `t between 0 and 1' appears to require our scalar domain to look like the real line. We can replace the condition's use of t and (1-t) with scalars a and b constrained to have sum 1: in a positive scalar domain, this does what we want. Otherwise, we shall need to impose some condition of being `positive' and `real'.

The constraining of mf(A) to lie in m(A).convexHull(A|f|) (ie the set {m(A).(pf(a) + qf(b)): a, b in A, p, q scalars with p+q=1}) only does what we wanted of integration if MD contains arbitrarily small neighbourhoods of arbitrary members of D and the convex hulls of f's images of these sets are arbitrarily small.

Written by Eddy.
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