# The Unapologetic Mathematician

## Product Measures

We continue as yesterday, considering the two $\sigma$-finite measure spaces $(X,\mathcal{S},\mu)$ and $(Y,\mathcal{T},\nu)$, and the product measure space $(X\times Y,\mathcal{S}\times\mathcal{T})$.

Last time we too a measurable set $E\subseteq X\times Y$ and defined the functions $f(x)=\nu(E_x)$ and $g(y)=\mu(E^y)$. We also showed that

$\displaystyle\int f\,d\mu=\int g\,d\nu$

That is, for every measurable $E$ we can define the real number

$\displaystyle\lambda(E)=\int\nu(E_x)\,d\mu=\int\mu(E^y)\,d\nu$

I say that this function $\lambda$ is itself a $\sigma$-finite measure, and that for any measurable rectangle $A\times B$ we have $\lambda(A\times B)=\mu(A)\nu(B)$. Since measurable rectangles generate the $\sigma$-ring $\mathcal{S}\times\mathcal{T}$, this latter condition specifies $\lambda$ uniquely.

To see that $\lambda$ is a measure, we must show that it is countably additive. If $\{E_n\}$ is a sequence of disjoint sets then we calculate

\displaystyle\begin{aligned}\lambda\left(\biguplus\limits_{n=1}^\infty E_n\right)&=\int\nu\left(\left(\biguplus\limits_{n=1}^\infty E_n\right)_x\right)\,d\mu\\&=\int\nu\left(\biguplus\limits_{n=1}^\infty(E_n)_x\right)\,d\mu\\&=\int\sum\limits_{n=1}^\infty\nu\left((E_n)_x\right)\,d\mu\\&=\sum\limits_{n=1}^\infty\int\nu\left((E_n)_x\right)\,d\mu\\&=\sum\limits_{n=1}^\infty\lambda(E_n)\end{aligned}

where we have used the monotone convergence theorem to exchange the sum and the integral.

We verify the $\sigma$-finiteness of $\lambda$ by covering each measurable set $E$ by countably many measurable rectangles with finite-measure sides. Since the sides’ measures are finite, the measure of the rectangle itself is the product of two finite numbers, and is thus finite.

We call the measure $\lambda$ the “product” of the measures $\mu$ and $\nu$, and we write $\lambda=\mu\times\nu$. We thus have a $\sigma$-finite measure space $(X\times Y,\mathcal{S}\times\mathcal{T},\mu\times\nu)$ that we call the “cartesian product” of the spaces $(X,\mathcal{S},\mu)$ and $(Y,\mathcal{T},\nu)$.

July 23, 2010 - Posted by | Analysis, Measure Theory

1. Professor Armstrong,

I think that Feynman’s path integrals of quantum field theory might provide the kind of product measures that we were discussing although I am not so sure how rigorous they are considered to be mathemnatically. I wished to email this note to you, but I could not figure out where to find your email. It is probably not posted.

Comment by Hamid | July 25, 2010 | Reply

2. They’re completely unrigorous, mathematically; that’s the whole issue with them! The path-integral is a wonderful heuristic, that can be interpreted in many cases to give a mathematically-sensible statement.

But in general there is no known coherent analogue of something like Lebesgue measure on function spaces — no way to “integrate over all paths” as the plain reading of the path-integral formulation suggests. Finding a universally-applicable interpretation of the path-integral heuristic instead of problem-specific ad hoc methods is the single greatest puzzle of mathematical physics.

Comment by John Armstrong | July 25, 2010 | Reply

3. […] of . Now if we have a measure on and Lebesgue measure on the Borel sets, we can define the product measure on . Since we know and are both measurable, we can investigate their measures. I assert […]

Pingback by The Measures of Ordinate Sets « The Unapologetic Mathematician | July 26, 2010 | Reply

4. […] and Iterated Integrals Let and be two -finite measure spaces, and let be the product measure on […]

Pingback by Double and Iterated Integrals « The Unapologetic Mathematician | July 27, 2010 | Reply

5. […] We continue our assumptions that and are both -finite measure spaces, and we consider the product space […]

Pingback by Fubini’s Theorem « The Unapologetic Mathematician | July 28, 2010 | Reply

6. I can suggest to see the following papers where can be found constructions of partial analogs of Lebesgue measure on some function spaces. There are

1) Baker, R., “Lebesgue measure” on $R\sp \infty$. Proc. Amer.
Math. Soc., 113(4) (1991), 1023–1029.

2) Baker, R., “Lebesgue measure” on $\Bbb R\sp \infty$. II.Proc. Amer. Math. Soc., 132(9), (2004), 2577–2591

3) Pantsulaia,G., On ordinary and Standard Lebesgue Measures on $R^{\infty}$, Bull. Polish Acad. Sci.73(3) (2009), 209-222.

4) Pantsulaia,G., On a standard product of an arbitrary family of sigma-finite Borel measures with domain in Polish spaces, Theory Stoch. Process, vol. 16(32), 2010, no 1, p.84-93.

Comment by Gogi Pantsulaia | December 17, 2010 | Reply