Consider a sequence $f_{n}: \mathbb{R} \rightarrow \mathbb{R}$ of measurable functions converging pointwise to a function $f: \mathbb{R} \rightarrow \mathbb{R}$. The Lebesgue measure is denoted by $\lambda$.

(a) Consider a Borel set $A \subset \mathbb{R}$ with finite Lebesgue measure $\lambda(A)<+\infty$. Define for $k, n \geqslant 1$ the sets

$$E_{n}^{(k)}:=\bigcap_{m \geqslant n}\left\{x \in A|| f_{m}(x)-f(x) \mid \leqslant \frac{1}{k}\right\}$$

Prove that for any $k, n \geqslant 1$, one has $E_{n}^{(k)} \subset E_{n+1}^{(k)}$ and $E_{n}^{(k+1)} \subset E_{n}^{(k)}$. Prove that for any $k \geqslant 1, A=\cup_{n \geqslant 1} E_{n}^{(k)}$.

(b) Consider a Borel set $A \subset \mathbb{R}$ with finite Lebesgue measure $\lambda(A)<+\infty$. Prove that for any $\varepsilon>0$, there is a Borel set $A_{\varepsilon} \subset A$ for which $\lambda\left(A \backslash A_{\varepsilon}\right) \leqslant \varepsilon$ and such that $f_{n}$ converges to $f$ uniformly on $A_{\varepsilon}$ as $n \rightarrow+\infty$. Is the latter still true when $\lambda(A)=+\infty$ ?

(c) Assume additionally that $f_{n} \in L^{p}(\mathbb{R})$ for some $p \in(1,+\infty]$, and there exists an $M \geqslant 0$ for which $\left\|f_{n}\right\|_{L^{p}(\mathbb{R})} \leqslant M$ for all $n \geqslant 1$. Prove that $f \in L^{p}(\mathbb{R})$.

(d) Let $f_{n}$ and $f$ be as in part (c). Consider a Borel set $A \subset \mathbb{R}$ with finite Lebesgue measure $\lambda(A)<+\infty$. Prove that $f_{n}, f$ are integrable on $A$ and $\int_{A} f_{n} d \lambda \rightarrow \int_{A} f d \lambda$ as $n \rightarrow \infty$. Deduce that $f_{n}$ converges weakly to $f$ in $L^{p}(\mathbb{R})$ when $p<+\infty$. Does the convergence have to be strong?