Fubini's Theorem - Corollary: product of integrals is the integral of the product.

Question

Let $\phi:[a,b]\rightarrow \mathbb{R}$ and $\psi:[a,b]\rightarrow \mathbb{R}$ integrable functions, then $f:A\rightarrow \mathbb{R}$ defined on $A=[a,b]\times [c,d]\subset \mathbb{R}^{2}$ for $f(x,y)=\phi(x)\psi(y)$ is integrable and $$\int_{A}f(x,y)dxdy=\bigg(\int_{a}^{b}\phi(x)dx\bigg)\bigg(\int_{c}^{d}\psi(y)dy\bigg).$$ I don't know how to proof the integrability of $f(x,y)$, proved this is enough to use the Fubini's theorem.

Someone can help me? any tips? Thanks !

Answer 1

Using Tonelli's theorem: $$\int_A |f(x, y)|dxdy = \left(\int_a^b|\phi(x)|dx\right)\cdot\left(\int_c^d |\psi(y)|dy \right)<\infty $$so that $f$ is integrable.