Projections, Radon-Nikodym, and conditioning

Transcription

1 Projections, Radon-Nikodym, and conditioning 1 Projections in L Radon-Nikodym theorem Conditioning Properties of K as a map from L 2 (Q) to L 2 (P ) S:projection 1 Projections in L 2 For a measure space (X, A, µ), the set L 2 = L 2 (X, A, µ) of all A\B(R)- measurable real functions on X would be a Hilbert space if we worked with equivalence classes of functions that differ only on µ-negligible sets. The corresponding set L 2 (X, A, µ) can be identified as a true Hilbert space. Lazy probabilists (like me) often ignore the distinction between L 2 and L 2, referring to f 2 = ( µ(f 2 ) ) 1/2 as a norm on L 2 (rather than using the more precise term semi-norm) and f, g = µ(fg) for f, g L 2 (X, A, µ) as an inner product. It is true that f, g is linear in f for fixed g and linear in g for fixed f, and it is true that f 2 = f, f, but we can only deduce that f(x) = 0 a.e.[µ] if f 2 = 0. As show by HW3.2, the space L 2 is also complete: for each Cauchy sequence {h n : n N} in L 2 there exists an h in L 2 (unique up to µ-equivalence) for which h n h 2 0. To avoid some tedious qualifications I will slightly abuse topological terminology by referring to a subset H of L 2 as closed to mean: if h n f 2 version: printed: 6 March 2017 c David Pollard

2 0 for a sequence {h n } in H and an f in L 2 then there exists an h in H for which h(x) = f(x) a.e.[µ]. (Some authors would insist that f itself should belong to H.) The following key result underlies the existence of both Radon-Nikodym derivatives (densities) for measures and Kolmogorov conditional expectations. projection <1> Theorem. Suppose H is a closed subspace of L 2 (X, A, µ). For each f L 2 there exists an f 0 in H for which (i) f f 0 2 = δ := inf{ f h 2 : h H} (ii) f f 0, h = 0 for every h in H (iii) property (iii) uniquely determines f 0 up to µ-equivalence. Proof The argument uses completeness of L 2 and the identity parallel <2> a + b a b 2 2 = 2 a b 2 2 for all a, b L 2, an equality that results from expanding a + b, a + b + a b, a b then cancelling out a, b terms. By definition of the infimum, for each n N there exists an h n H for which f h n 2 δ n := δ + n 1. Invoke equality <2> with a = f h n and b = f h m, simplifying a + b 2 2 to 4 f (h n + h m )/2 2 2 then rearranging: to give 4δ 2 4 f (h n + h m )/ h n h m 2 2 = 2 f h n f h m 2 2 h n h m 2 2 2δ2 n + 2δ 2 m 2δ 2 0 as min(m, n). That is {h n } is a Cauchy sequence, which converges in norm to an f 0 in L 2. Without loss of generality we may assume that f 0 H. By construction, δ f f 0 2 f h n 2 + h n f 0 2 δ as n. Equality (i) follows. Draft: c David Pollard 2

3 For (ii) note, for each h H, that the quadratic f (f 0 + th) 2 = f f t f f 0, h + t 2 h 2 2 achieves its minimum value δ 2 at t = 0, which forces the coeffiecient of t to equal zero. If f 0, f 1 H and both f f 0 and f f 1 are orthogonal to each h in H then the difference f 0 f 1 must be orthogonal to itself, that is f 0 f = 0, forcing f 0 = f 1 a.e.[µ]. The function f 0 is called the (orthogonal) projection of f onto H. We may regard the projection function π H, which takes the f from Theorem <1> to f 0, as a map from L 2 into µ-equivalence classes of functions in H. If we arbitrarily choose one member from each equivalence class then π H can also be thought of as a map from L 2 into H. If g 1, g 2 L 2 and π H g i = h i for i = 1, 2 then, for constants c 1 and c 2, c 1 (g 1 h 1 )+c 2 (g 2 h 2 ), h = c 1 g 1 h 1, h +c 2 g 2 h 2, h = 0 for h H. By part (iii) of the Theorem, c 1 h 1 + c 2 h 2 is µ-equivalent to π H (c 1 g 1 + c 2 g 2 ). That is, ae.lin <3> π H (c 1 g 1 + c 2 g 2 ) = c 1 π H g 1 + c 2 π H g 2 a.e.[µ], which is as close to linearity as we can hope to get for a map that is only defined up to a µ-equivalence. S:RN 2 Radon-Nikodym theorem The simplest form of the theorem concerns two finite measures µ and ν defined on some (X, A). RN.simple <4> Theorem. If µx < and µf νf for each f in M + (X, A) then there exists an A-measurable function with 0 (x) 1 for all x such that νf = µ(f ) for each f in M + (X, A). The function is unique up to µ-equivalence. Proof (sketch of a proof due to von Neumann, 1940, page 127) Without loss of generality suppose νx = 1. Draft: c David Pollard 3

4 Define H = {f L 2 (X, A, µ) : νf = 0}. Note that H is a closed subspace of L 2 : if νh n = 0 and µ h n f 2 0 then νf = ν(h n f) ν h n f ν h n f 2 µ h n f 2 0, implying νf = 0. Let f 0 = π H 1 and g = 1 f 0. Note that νg = ν1 νf 0 = 1. In consequence µ{x : g(x) 0} ν{x : g(x) 0} > 0 and g 2 2 = µ(g2 ) > 0. Suppose f L 2 has νf = c. Then ν(f cg) = 0, that is, f cg H, so that 0 = f cg, g = µ(fg cg 2 ) = 0. The final equality rearranges to νf = µ(f ) where := g/ g 2 2. Invoke the last equality with f = {δ < 0} to get 0 ν{ < 0} µ ( { < 0}) 0, with the last inequality strict unless µ{ < 0} = 0. Similarly µ{ > 1} ν{ > 1} = µ ( { > 1}) µ{{ > 1}, with the last inequality strict unless µ{ > 1} = 0. Replace by {0 1}. For f M + (X, A) take limits (MC) in ν(f n) = µ (f n) as n. If µ(f 1 ) = µ(f 2 ) for all f M + consider first f = { 1 < 2 } then f = { 1 > 2 } to deduce that 1 = 2 a.e.[µ]. Theorem <4> has an extension to sigma-finite measures with ν dominated by µ, that is, if νa = 0 if A A and µa = 0. RN.sigma.finite <5> Theorem. If µ and ν are both sigma-finite measures with ν dominated by µ then there exists a real-valued function M + (X, A) for which νf = µ(f ) for each f in M + (X, A). The function is unique up to µ-equivalence. For an idea of the proof see Pollard (2001, Section 3.2). Draft: c David Pollard 4

5 S:Kolmogorov 3 Conditioning Recall the conditioning problem. We have a probability measure Q on the product sigma-field A B on X Y, with X-marginal P. That is, P g = Q x,y g(x) for each g M + (X, A). Here and subsequently I identify g with a function on X Y whose value does not depend on the Y-coordinate. We seek a Markov kernel K = {K x : x X} a family of probability measures on B for which x K x B is A-measurable for each B B for which disint <6> Qf(x, y) = P x K y xf(x, y) for each f M + (X Y, A B). The measure K x s needs to have the linear functional properties that identify it as an integral with respect to a probability measure on B. That is, K x : M + (Y, B) [0, ] and (i) K x 0 = 0 and K x 1 = 1; (ii) K x (c 1 f 1 + c 2 f 2 ) = c 1 K x f 1 + c 2 K x f 2 for constants c i R + ; (iii) K x f 1 K x f 2 if f 1 (y) f 2 (y) for all y; (iv) if f n (x, y) f(x, y) then K y xf n (x, y) K y xf(x, y). As you will soon see, the properties of projections will provide a map K x : M + (Y, B) [0, ] for which analogous properties hold except on various Q negligible sets. In actually takes a lot of work to collect up these (uncountably many) Q-negligible sets into a single negligible set, outside of which the K x behaves like a Markov kernel. For many purposes, the unpleasant task of cleaning up bad behavior on negligible sets can be ignored if we are content to work with an object K = {K x : x X} that shares many of the desirable properties of Markov kernels without actually defining integrals with respect to probability measures. That object is usually known as a Kolmogorov conditional expectation operator. Here how projections get into the story. We can identify L 2 (P ) := L 2 (X, A, P ) with a subspace of L 2 (Q) := L 2 (X Y, A B, Q) because P g(x) 2 = Qg(x) 2. In fact L 2 (P ) is a closed subspace of L 2 (Q). Indeed, if {g n } is a sequence in L 2 (P ) for which Q g n (x) f(x, y) 2 0 for some f L 2 (Q). Draft: c David Pollard 5

6 By an argument similar to HW2.2 and HW3.2, there exists a subsequence along which g n(k) (x) f(x, y) a.e.[q]. Deduce that f is Q-equivalent to lim sup n(k) g n (k)(x). Minor surgery on some P -negligible sets excludes values where the limsup equals ±, leaving a function in L 2 (P ). For an f L 2 (Q) define g(x) = K y xf(x, y), an A-measurable function for which, by Jesnsen s inequality, P g 2 P (K y xf(x, y) 2 ) Qf 2 <. Moreover, for each h L 2 (P ), Q (f(x, y) g(x)) h(x) = P x (h(x)k y x(f(x, y) g(x)) = 0. That is, f g is orthogonal to H in the sense of the L 2 (Q) inner product, the property that identifies g(x) as one of the L 2 (P ) functions that represents the orthogonal projection of f onto L 2 (P ). Let me write K x f for some arbitrarily choice for the projection of f onto L 2 (P ), with the subscript x to remind us that the projection is a function of x. No prizes for guessing why I choose to write the function in this strange form. Don t get overexcited and write K x instead of K x. 3.1 Properties of K as a map from L 2 (Q) to L 2 (P ) To be continued. References PollardUGMTP vonneumann1940annmath Pollard, D. (2001). A User s Guide to Measure Theoretic Probability. Cambridge University Press. von Neumann, J. (1940). On rings of operators. III. Annals of Mathematics 41 (1), Draft: c David Pollard 6