Indicator and simple functions verify Fubini's Theorem (Lebesgue-Integral)

Question

I'm working on a proof of Fubini's Theorem. The theorem says:

Given $A\times B\in \mathcal{L}\times\mathcal{L}$ a Lebesgue measurable set in $\mathbb{R^2}$, and $f:A\times > B\rightarrow\overline{\mathbb{R}}$ a Lebesgue integrable function, then: $$\iint_{A\times B}f(x,y)dxdy=\int_A(\int_Bf(x,y)dy)dx=\int_B(\int_Af(x,y)dx)dy.$$

My proof consists in four steps:

1. Prove it for indicator functions $\chi_E$, being $E\in\mathcal{L}\times\mathcal{L}.$
2. Prove it for simple functions $S$.
3. Prove it for Lebesgue measurable positive functions $f$.
4. Extend it to real measurable functions $f$, using the decomposition $f=f^+-f^-$.

I've already proven points 3 and 4 assuming the first two. My questions is about proving it for cases 1 and 2. I will thank any help.

Answer 1

You should include "step $0$", which is to prove it for indicator functions of the form $\chi_{A\times B}$, where $A, B\in \mathcal{L}$. Then use the fact that these kind of sets, generate your $\sigma$-algebra