I understood the proof of this theorem completely, except the line which I've underlined. Intuitively I understand this but I am not able to write down rigorous proof.
If we have some measurable function $f:X\to [-\infty,+\infty]$ then we can approximate it with simple function namely $\{\varphi_k\}_{k\geq 1}$. And they are approximating each this function $\varphi_k$ with sequence of step functions. Finally, we get sequence of sequences and I am get confused by this.
Would be very grateful if anyone can explain me this in rigorous way!
Indeed, this is not completely straightforward, I prove this when $f(x)$ is bounded, the general case is an easy modification.
The point is that to every $k$ there is a simple function $g_k$ such that $|f-g_k|< 1/2^k$ uniformly, which follows from the usual construction of $g_k$. By the argument above there is a step function $s_k$ such that $m(E_k)<1/2^k$, where $$E_k=\{x: |g_k(x)-s_k(s)|>0\}.$$
Then also $m(F_k)<1/2^k$, where $F_k=\{x: |f(x)-s_k(x)|> 1/2^k\}$. Since $\sum_k m(E_k) < 1$ by the Borel-Cantelli Lemma we have that for a.e. $x$, the point $x$ is not in $E_k$ for all $k>N$. Thus $|f(x)-s_k(x)|\le 1/2^k$ for all $k>N$ and hence $s_k(x) \to f(x)$.