# Calculating a product measure with Fubini-Tonelli's theorem

I am kind of lost on the following problem. Let $${f : (X, \mathcal{A}, \mu) \rightarrow (\mathbb{R_{\geq}}, \mathcal{B}(\mathbb{R_{\geq}}), \lambda)}$$ with $${\lambda}$$ being the Lebesgue-measure.

1) Show that $${E \in \mathcal{A} \otimes \mathcal{B}(\mathbb{R_{\geq}})}$$ with $${E:= \{(x,y) \in X \times \mathbb{R_{\geq}} | x \in X \wedge 0 \leq y < f(x) \}}$$

For this, my approach was $${\mathcal{A} \otimes \mathcal{B}(\mathbb{R_{\geq}}) = \sigma(\mathcal{A} \times \mathcal{B}(\mathbb{R_{\geq}}) \cup \mathcal{B}(\mathbb{R_{\geq}}) \times \mathcal{A})}$$ $${X \times \mathbb{R_{\geq}} \subset \mathcal{A} \times \mathcal{B}(\mathbb{R_{\geq}}) \subset \mathcal{A} \times \mathcal{B}(\mathbb{R_{\geq}}) \cup \mathcal{B}(\mathbb{R_{\geq}}) \times \mathcal{A} \subset \mathcal{A} \otimes \mathcal{B}(\mathbb{R_{\geq}})}$$

at which point, if it is correct, this would be proven. Now the second part is where I am really stuck:

2) Calculate $${\mu \otimes \lambda (E)}$$ using Tonelli's theorem, to prove the following equation: $${\int f d\mu = \int_0^\infty \mu(\{f > y\} \lambda(dy))}$$

I cannot comprehend how to solve $${\mu \otimes \lambda (E)}$$, since in my understanding it would just be something like $${\mu \otimes \lambda (E) = \mu(E_1) * \lambda(E_2)}$$, with $${E_1 \in X}$$ and $${E_2 \in \mathbb{R_\geq}}$$.

So I started by rewriting the equation to $${\int_0^\infty \mu(\{f> y\}) \lambda(dy) = \int_0^\infty(\int_0^y \chi_{\{ f > y\}} \mu(dx)) \lambda(dy)}$$ so that I can switch the integrals and end up with $${\int_0^\infty \lambda(\{f>y\}) \mu(dx)}$$ Is this a correct approach? Because I cannot see it going anywhere near the needed solution and I do not know how else to apply the theorem of Tonelli/Fubini...

## 2 Answers

First part is wrong. write $$E$$ as $$\cup_{r\in \mathbb Q} (f>r) \times ([0,r)$$ to see that $$E$$ belongs to the product sigma algebra. For the second part you have started correctly. Just note that $$\lambda (\{f(x)>y\})$$ is nothing but the Lebesgue measure of the interval $$[0,f(x))$$ which is $$f(x)$$.

• Thank you for your answer, the second part is clear to me now. However, I don't quite get the first part yet: did you mean rewrite ${E}$ as ${\cup_{r \in \mathbb{Q}}(\{ f > r \} \cap [0,r))}$? I am sorry if it's a dumb question, but I can't quite comprehend how that works. ${E}$ is defined to consist of pairs ${(x,y)}$, would that structure not be destroyed when rewriting it like you suggested? – NotAName Jan 28 at 23:53
• @NotAName Sorry, I typed $\cap$ instead of $\times$. – Kavi Rama Murthy Jan 28 at 23:57

The other answer covers it, but here is a somewhat different approach.

To do the first part, we use a trick: set $$\alpha(x,y)=(f(x),y)$$ and $$\beta (x,y)=x-y$$. Then $$\beta\circ \alpha(x,y)=f(x)-y.$$ Now, $$\beta$$ is obviously continuous and it's not hard to show that $$\alpha$$ is measurable. Then, $$E=\{\beta\circ \alpha>0\}$$ is measurable.

And for the second part, we can use a result that does not use the full strength of Tonelli. Namely,

$$\mu\times \lambda(E)=\int_X\lambda(E_x)d\mu=\int_Xf(x)d\mu=\int^{\infty}_0\mu(E^y)d\lambda=\int^{\infty}_0\mu(\{f>y\})d\lambda$$

• Thanks for the answer. I like the trick, but I have a question concerning the second part, although it might just be the notation which confuses me. Why did you write the ${x}$ in ${\lambda(E_x)}$ in the second step? Is that just to show that you integrate over ${X}$? And why is the ${y}$ in the later step on the upper side (${E^y}$)? Can these two be left out or do they have a special meaning? – NotAName Jan 29 at 0:29
• $E_x=\{y:(x,y)\in E\}$. It is a "section" of $E$. Notice, $\lambda(E_x)$ defines a function of $x$, which you are integrating over $\mu$. The idea is that you fix (momentarily) $x$ , calculate the (Lebesgue) measure of the strip obtained, which in our case, is $f(x)-0=f(x)$. Now let $x$ vary, and integrate this function with respect to $\mu$. Similarly, for $E^y$. The positions of $x$ and $y$ is just standard notation. If you go through the proof of Tonelli/Fubini, you will see these formulas as precursors to the main theorem. Folland has a nice treatment of this. – Matematleta Jan 29 at 0:38