About the Lebesgue integrability of $f$

Question

I'm trying to solve the following problem:

Let $f \ge 0$ be a measurable function on a measurable subset $E \subset \mathbb{R}^d$. Suppose there exits a sequence of measurable subset $\{E_k\}_{k=1}^{\infty}$ of $E$ such that $$m(E\setminus E_k)<\frac{1}{k}\quad \textrm{ and }\quad \lim_{k\to \infty}\int_{E_k} f(x)dx<\infty.$$ Show that $f$ is integrable on $E$.

proof:

Consider $$ f_k:=\chi_{E_k}f \Longrightarrow \lim_{k\to \infty}f_k=f \textrm{ a.e on }E \quad \textrm{and}\quad f_k\ge 0. $$ We can see \begin{align*} \int_{E}f dm&=\int_{E}\left(\liminf_{k \to \infty} f_k \right)dm \leq \liminf_{k \to \infty}\int_{E}f_k dm \quad \textrm{by Fatou's Lemma} \\  =&\liminf_{k \to \infty}\int_E \chi_{E_k}f dm\leq \lim_{k \to \infty} \int_{E_k}f dm <\infty \quad \textrm{by hypothesis} \end{align*}

Thanks for any hint.

Answer 1

Hint: apply Fatou's lemma to the sequence of functions $f_k = f 1_{E_k}$.