Product Measures and Fubini's Theorem

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Douglas Brooks
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1 Product Measures and Fubini's Theorem 1. Product Measures Recall: Borel sets U in are generated by open sets. They are also generated by rectangles VœN " á N hich are products of intervals NÞ 3 Let V be the collection of all rectangles - e have shon that U œ ÐVÑ, i.e. the Borel sets are the smallest -field containing all rectangles in. No e distinguish rectangles (products of intervals in " ) from measureable rectangles (products of arbitrary Borel sets in " Ñ Definition 1: The measureable rectangles W in are the collection of products of Borel sets: W œöf á F ÀF " 3 are Borel in We kno that Borel sets are V Ð Ñ. But e can sho they are also W Ð Ñ. Lemma 1: The Borel sets are the same as W Ð Ñ. Proof: We need to sho that W Ð Ñis contained in Borel sets U. If N3 are open intervals, then N " N# á N U, and further the collection ÖEÀE N # á N U is easily seen to be a -field (hich contains intervals), and so must be U. Thus for F" U, it follos that F " N# á N U if N3 are open intervals. A similar agument shos that for F U and N are open intervals, then # 3 F F N á N " # $ 8 U. Continuing this ay, e conclude that F " á F for Borel F3, shoing that W Ð Ñ U. This shos that W Ð Ñ œ U. Recall Lebesgue measure 7 on is defined by extending it from the rectangles V uniquely. It is no easily shon that 7 can also be extended U 99

2 from the measureable rectangles W uniquely, since for F 3 U, it is easy to sho that 7ÐF á F Ñ œ 7ÐF Ñá 7ÐF Ñ " " Note that products of measures. 3 on can be defined analogously to Lebesgue measure on products of copies of the real line. Specifically, if." is a Borel measure on.# is a Borel measure on ã. is a Borel measure on Then e can define the as follos: product measure. œ.".# á. on Recall œöðb ßáßB ÑÀ B. " 3 Let F" U œ Borel sets on F # U ã F U We can similarly define. ÐF " á F Ñ œ." ÐF" Ñá. ÐFÑÞ and then extend to all Borels U Þ Thus: Theorem 2: b unique measure. on defined by.... ÐF F á F Ñœ ÐF Ñ ÐF Ñá ÐF Ñ " # " " # # 100

3 Of course, this definition generalizes to any product of measure spaces Ð\ß k ß. Ñ Ð] ß l ß / Ñ H œ k l œ ÖÐBßCÑ À B \ß C ]Ñ Thus: on F l. He define a -field Y œ k l œ ÖE F À E kß Measure 1 œ. / as before is defined by 7ÐE FÑ œ. ÐEÑ / ÐFÑÞ Theorem 18.1 [Billingsley]: (a) If I k l, then for each B, ÖC À ÐBß CÑ I l, and similarly for eacy C, ÖB À ÐBß CÑ I kþ (b) If 0ÐBßCÑ measureable rt k l, then for each fixed B 0ÐBß Ñ is measureable ith respect to l, and for each fixed C0Ð ßCÑis measurable ith respect to k. Proof: Given fixed B the map XBÐCÑ œ ÐBß CÑ maps ] to \ ]. Easy to " sho that X ÐIÑis measureable if I œ E F is a measureable rectangle. Thus by Theorem 13.1 X is measurable since rectangles I generate all measureable sets. Thus for any I e have XB " ÐIÑ œ ÖC À ÐBßCÑ I is measureable. The final claim follos easily. [simple proof in Billingsley] No consider a set I k l. Define IB œxb " ÐIÑ. Define the function 7ÐBÑ œ / ÐIB Ñ œ / ÖC À ÐBßCÑ I. It is easy to sho that 7ÐBÑ is measureable (see Billingsley). Define 1 ÐIÑ œ ( / ÐIBÑ.. ÐBÑ \ here 1 ÐIÑ œ (. ÐICÑ. / ÐCÑ ] 101

4 IC œ ÖB À ÐBßCÑ I. Easy to sho that 1 and 1 are measures. Also easy to sho that if IœE F is a measureable rectangle then 1 ÐIÑ œ. ÐEÑ/ ÐFÑ œ 1 ÐIÑ œ 1 ÐIÑÞ Using unique extension (see Billingsley; this must be done first for finite and then -finite measure spaces), e obtain: Theorem 18.2 [B]: The measure 1 œð. / Ñ 1 ÐE FÑ œ. ÐEÑ/ ÐFÑ defined by is -finite and conicides ith 1 and 1 above. It is its on unique extension hen restricted to measureable rectangles IœE F. Example 1: Lebesgue measure on is. =.".# here. 3 are Lebesgue measure. # 2. The Fubini theorem (analysis; section 18) Fubini Theorem (section 18): If 0ÐBßCÑ is measurable ith respect to.., and ' l0l.ð.. Ñ _, then " # " # ( Œ( 0ÐBß CÑ.. ÐBÑ.. ÐCÑ œ ( 0ÐBß CÑ.Ð.. Ñ " # " # œ ( Œ( 0ÐBß CÑ..# ÐCÑ.." ÐBÑÞ This is effectively a fundamental theorem of calculus (an evaluation theorem) for multiple integrals. Proof: In the case here 0ÐBßCÑ œ MIÐBßCÑ here I k l, this is easy to sho (follos directly from Theorem 18.2 above). On the other 102

5 hand, this then extends to simple functions and then by taking limits directly to positive functions, and then all functions. 103

Notes on the Unique Extension Theorem. Recall that we were interested in defining a general measure of a size of a set on Ò!ß "Ó.

1. More on measures: Notes on the Unique Extension Theorem Recall that we were interested in defining a general measure of a size of a set on Ò!ß "Ó. Defined this measure T. Defined TÐ ( +ß, ) Ñ œ, +Þ