I am new to solving non trivial-for me,at least- theorical problems,so I Have kind of general trouble, when constructing a proof, to see if all passages are correct. In particular, I tried to prove the following statement and I would need either a check to my solution or a new one if my attempt is not correct.
Let $(f_h) \subset L^1(\mathbb{R}^2)$ a sequence of functions converging in $L^1(\mathbb{R}^2)$ to $0$, i.e.
$\lim_{n \to \infty}\int_{\mathbb{R}^2}|f_n(x,y)|\,dxdy=0$
Prove there exists a subsequence $f_{h(k)}$ such that, for almost everywhere $x \in \mathbb{R}$, the sequence
$g_{h(k)}^x(y)= f_{h(k)}(x,y) \;$
converges to $0$ in $L^1(\mathbb{R})$ ,which is $\lim_{k \to \infty} \int_{\mathbb{R}}|f_{h(k)}(x,y)|\, dy=0 $ for almost anywhere $x \in \mathbb{R}$
Here my attempt:
By fubini theorem , we get that:
$\int_{\mathbb{R}^2}|f_n(x,y)|\,dxdy= \int_{\mathbb{R}}(\int_{\mathbb{R}}|f_n(x,y)|\,dy)\,dx \to 0$
as $n$ approaches to infinity. This is equivalent to say that the sequence of function of $x$ ,
$w_n(x)= \int_{\mathbb{R}} |f_n(x,y)|\, dy$
tends to $0$ in $L^1(\mathbb{R})$. By a well known corollary to Riestz-Fisher theorem there exists a subsequence $w_{h(k)}$ such that
$\lim_{k \to \infty} w_{h(k)}(x)=0 \qquad $ for almost anywhere $x \in \mathbb{R} $
which is equivalent to
$\lim_{k \to \infty} \int_{\mathbb{R}}|f_{h(k)}(x,y)|\, dy = 0 \qquad$ for almost anywhere $x \in \mathbb{R}$
which is the desired result.