I am studying the generalized minkowski inequality for integrals:
Let be $E\subset \mathbb{R^n}$, $F\subset \mathbb{R^m}$, $1\leq p \leq \infty$, $f$ measurable in $E\times F$. Then
$$\left\| \displaystyle \int_{F}f(\cdot,y)dy\right\|_{L_{p,x}(E)}\leq \displaystyle \int_{F}\|f(\cdot,y)\|_{L_{p,x}(E)}dy$$.
The proof is the following,
Let be $\varphi(x)=\displaystyle \int_{F}f(x,y)dy$. So, we have
$$\| \varphi\|_{p}=\underset{g\in B_{p'}}{sup}\left| \displaystyle\int_{E}\varphi(x)g(x)dx\right|\leq\underset{g\in B_{p'}}{sup}\displaystyle\int_{E}\left(\displaystyle \int_{F}|f(x,y)||g(x)|dy\right)dx$$
The notes I am reading indicate that Fubini's theorem holds that
$$\displaystyle\int_{E}\left(\displaystyle\int_{F}|f(x,y)||g(x)|dy\right)dx=\displaystyle\int_{F}\left(\displaystyle \int_{E}|f(x,y)||g(x)|dx\right)dy$$
Fubuni's theorem asks that the function $f(x,y)g(x)$ to be integrable in $E\times F$, but with the given hypotheses I don't know how to guarantee this fact
Could someone explain to me how I can guarantee the application of Fubini's theorem?