# Show that $\frac{d(\alpha\times \beta)}{d( \mu\times \nu)}(x,y)=\frac{d\alpha}{d\mu}(x)\frac{d\beta}{d\nu}(y)$

Suppose $$(X,\mathcal M,\alpha)$$ and $$(Y,\mathcal N,\beta)$$ are positive finite measure spaces and $$(X,\mathcal M,\mu)$$ and $$(Y,\mathcal N,\nu)$$ are positive $$\sigma$$-finite measure spaces with $$\alpha\ll \mu$$ and $$\beta\ll \nu$$. Prove that $$\alpha\times \beta \ll \mu\times \nu$$ and $$\frac{d(\alpha\times \beta)}{d( \mu\times \nu)}(x,y)=\frac{d\alpha}{d\mu}(x)\frac{d\beta}{d\nu}(y)$$ for a.e. $$(x,y)\in X\times Y$$.

I think showing that $$\alpha\times \beta \ll \mu\times \nu$$ will be an application of the radon-nikodym theorem, but I don't see how exactly. I've been struggling with this problem for over a week, if somebody could hold me hand on this one and walk me through it you'd be a lifesaver!!

• Go through what I've written. If I'm a lifesaver, I'd like feedback. Apr 27, 2020 at 16:13
• Will do soon man, Got some stuff to do, i'll look at it later tho
– user637978
Apr 27, 2020 at 18:11

Showing $$(\alpha \times \beta) \ll (\mu \times \nu)$$ looks difficult.

After all, how would you show something like $$(\alpha \times \beta)A = \int_{A} \frac{d \alpha}{d \mu} (x)\frac{d \beta}{d \nu}(y)(\mu \times \nu)(x,y) \tag{1}$$ for every set $$A$$ in $$M \otimes N$$, the product sigma-algebra? It seems difficult because the nature of $$A$$ is not ascertainable from its membership in the product sigma-algebra, so we can't really simplify both sides of the equation.

The idea is to show that $$\mathcal S = \{A \in M \otimes N : (1) \text{ holds for } A\}$$ is a sigma-algebra which contains a generating set of $$M \otimes N$$.

What would the generating set be? Quite naturally, it would be the set of all rectangles in $$M \times N$$. That is: $$\mathcal T = \left\{A \times B : A \in \mathcal M, B \in \mathcal N \right\}$$

therefore, it is enough to show that for any $$A \in \mathcal T$$, the identity $$1$$ holds, and that $$\mathcal S$$ is a sigma-algebra.

Why does $$(1)$$ hold for $$A \in \mathcal S$$? Use the definition of the product measure. On any set of the form $$A \times B$$, $$(\alpha \times \beta)(A \times B) = \alpha(A) \times \beta(B) = \int_{A} \frac{d \alpha}{d \mu}(x)dx \int_B \frac{d \beta}{d \nu}(y)dy$$

Now, by definition of the product measure, the integral on the RHS of $$(1)$$ also simplifies. Recall that the integral of a function over a set in a product sigma-algebra is given by the integral of the slice functions (can take either variable first by Fubini's theorem). That is, if $$f : X \times Y \to \mathbb R$$ is integrable, then : $$\int_{X \times Y} fd(\alpha \times \beta)(x,y) = \int_X \left(\int_Y f_x d\beta(y)\right) d \alpha(x) = \int_{Y} \left(\int_X f_x d \alpha(x)\right)d\beta(y)$$

where $$f_x(y) = f(x,y) : Y \to \mathbb R$$, similarly $$f_x$$.

With that, observe that if a function $$h(x,y)$$ is of the form $$h(x,y) = f(x)g(y)$$, then : $$\int_Y h_x d \beta(y) = f(x)\int_Y g(y) d \beta(y)$$ where $$\int_Y g(y) d \beta(y)$$ is a constant independent of $$x$$, so it can be slipped out of any integral involving $$x$$. Therefore, using this to simplify the RHS of $$1$$ : $$\int_{A \times B} \frac{d \alpha}{d \mu}(x) \frac{d \beta}{d \nu}(y) (\mu \times \nu)(x,y) = \int_A \frac{d \alpha}{d \mu}(x)d \mu(x) \int_B \frac{d \beta}{d \nu} d \nu(y)$$

which proves $$1$$ for $$A \times B$$.

Next, we show that $$\mathcal S$$ is a sigma-algebra. For this, we employ the Dynkin $$\pi-\lambda$$ theorem. To show this , we need a $$\pi$$ system which generates $$M \otimes N$$, and need to show that $$\mathcal S$$ is a $$\lambda$$ system.

The $$\pi$$ system is straightforward enough : the set of all finite disjoint unions of sets from $$\mathcal T$$ is a $$\pi$$ system (one can easily check this). That $$\mathcal S$$ contains this set follows from the fact that the product is integral is finitely additive.

Now, $$\mathcal S$$ is a $$\lambda$$ system needs to be checked. Check the conditions on Wikipedia for yourself( the second set of conditions) and see that they are satisfied (it is fairly clear).

$$(1)$$ clearly implies that $$\alpha \times \beta \ll \mu \times \nu$$.

Therefore, $$(1)$$ holds on $$M \otimes N$$. Because the RHS should also be $$\int_A \frac{d(\alpha \times \beta)}{d (\nu \times \mu)}(x,y) (\mu \times \nu)(x,y)$$, from $$(1)$$ holding for all such subsets we get by uniqueness of the RN derivative , the equality a.e. of the required functions.