Product Measures and Fubini's Theorem

1. Product Measures

Recall: Borel sets U‘55 in are generated by open sets. They are also generated by rectangles VœN"5 ‚á‚N which are products of intervals NÞ3

Let VU5V be the collection of all rectangles - we have shown that 5 œÐÑ , i.e. the Borel sets are the smallest 5‘ -field containing all rectangles in 5 .

Now we distinguish rectangles (products of intervals in ‘" ) from measureable rectangles (products of arbitrary Borel sets in ‘"Ñ

Definition 1: The measureable rectangles W‘ in 5 are the collection of products of Borel sets:

W œÖF"53 ‚á‚F ÀF are Borel in ‘×

We know that Borel sets are 5VÐÑ . But we can show they are also 5W Ð Ñ .

Lemma 1: The Borel sets are the same as 5WÐÑ .

5 Proof: We need to show that 5WÐÑ is contained in Borel sets U . If N3 are 5 open intervals, then N‚N‚á‚N−"# 5U , and further the collection 5 ÖEÀE‚N#5 ‚á ‚N −U5 × is easily seen to be a -field (which contains intervals), and so must be UU . Thus for F−" , it follows that 5 F‚N‚á‚N−"# 5U if N 3 are open intervals. A similar agument shows that for F−#3U and N are open intervals, then 5 F‚F‚N‚á‚N−"#$ 8U . 5 Continuing this way, we conclude that F‚á‚F−"5U for Borel F 3 , showing that 5WÐѧ U5 .

This shows that 5WÐÑœ U5 .

Recall Lebesgue measure 7 on ‘5 is defined by extending it from the rectangles V uniquely. It is now easily shown that 7 can also be extended

99 from the measureable rectangles WU uniquely, since for F−3 , it is easy to show that

7ÐF"5"5 ‚ 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 .‘5 is a Borel measure on

Then we can define the product measure ..œ‚‚á‚"# . . 5 on ‘5 as follows:

5 Recall ‘‘œÖÐB"53 ßáßB ÑÀ B − × .

Let F−" U‘ œ Borel sets on F−# U

ã

F−5 U

We can similarly define

...ÐF"5""55 ‚ á ‚ F Ñ œ ÐF Ñá ÐF ÑÞ and then extend to all Borels U5Þ

Thus:

Theorem 2: b unique measure .‘ on 5 defined by

....ÐF"# ‚F ‚á ‚F 5 Ñœ ""##55 ÐFÑ ÐFÑá ÐFÑ

100 Of course, this definition generalizes to any product of measure spaces Ð\ßk. ß Ñ ‚ Ð] ß l/ ß Ñ

Hklœ‚œÖÐBßCÑÀB−\ßC−]Ñ× Thus: on H5Ykl5k we define a -field œ‚œÖE‚FÀE−ß F−l ×.

Measure 1./œ‚ as before is defined by 7ÐE ‚ FÑ œ./ ÐEÑ ‚ ÐFÑÞ

Theorem 18.1 [Billingsley]: (a) If I−kl ‚ , then for each B , ÖC À ÐBß CÑ − I× −lk, and similarly for eacy C , ÖB À ÐBß CÑ − I× − Þ (b) If 0ÐBßCÑ measureable wrt kl ‚ , then for each fixed B 0ÐBß †Ñ is measureable with respect to l , and for each fixed C0ІßCÑ is measurable with respect to k .

Proof: Given fixed B the map XB ÐCÑ œ ÐBß CÑ maps ] to \ ‚ ] . Easy to show that XÐIÑ" is measureable if IœE‚F is a measureable rectangle. Thus by Theorem 13.1 XI is measurable since rectangles generate all " measureable sets. Thus for any IXÐIÑœÖCÀÐBßCÑ−I× we have B is measureable. The final claim follows easily.

[simple proof in Billingsley]

" Now consider a set I−kl ‚ . Define IB œXB ÐIÑ . Define the function 7ÐBÑœÐIÑœÖCÀÐBßCÑ−I×//B . It is easy to show that 7ÐBÑ is measureable (see Billingsley). Define

w 1/.ÐIÑ œ ÐIB Ñ . ÐBÑ (\

ww 1./ÐIÑ œ ÐIC Ñ. ÐCÑ (] where

101 IœÖBÀÐBßCÑ−I×C .

Easy to show that 11www and are measures. Also easy to show that if IœE‚F is a measureable rectangle then 1./1ÐIÑ œ ÐEÑ ÐFÑ œwww ÐIÑ œ 1 ÐIÑÞ

Using unique extension (see Billingsley; this must be done first for finite and then 5 -finite measure spaces), we obtain:

Theorem 18.2 [B]: The measure 1./œÐ ‚ Ñ defined by 1./ÐE ‚ FÑ œ ÐEÑ ÐFÑ is 511 -finite and conicides with www and above. It is its own unique extension when restricted to measureable rectangles IœE‚F .

# Example 1: Lebesgue measure on ‘... is = "#‚ where . 3 are Lebesgue measure.

2. The Fubini theorem (analysis; section 18)

Fubini Theorem (section 18): If 0ÐBßCÑ is measurable with respect to ..‚l0l.ЂÑ_, and .. , then "#' "#

0ÐBß CÑ..."# ÐBÑ . ÐCÑ œ 0ÐBß CÑ.Ð .. " ‚# Ñ ((Œ (

œ 0ÐBß CÑ...#" ÐCÑ . ÐBÑÞ ((Œ This is effectively a fundamental theorem of calculus (an evaluation theorem) for multiple integrals.

Proof: In the case where 0ÐBßCÑ œ MI ÐBßCÑ where I −kl ‚ , this is easy to show (follows directly from Theorem 18.2 above). On the other

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

103