I'm trying to solve the following exercise:
Let $d$ be an integer $\geq 1$ and $f\in L^p(\mathbb{R}^d)$, $1 \leq p <\infty$. Let $B_k$ be the closed ball of $\mathbb{R}^d$ of centre $0$ and radius $k$, $k \in \mathbb{N}$.
Show that the sequence $ (\chi_{B_k}(f))_{k \in \mathbb{N}} $ converges in $ L^p(\mathbb{R}^d)$ to $f$.
I really have no idea on where to start. Should I try showing it on simple functions and use density theorems? Or maybe use a sequence $(f_n)_{k \in \mathbb{N}}$ that converges a.e. to $f$ and use triangle inequality or Hölder's?
I was thinking about a function $f_{\epsilon}\in C_C(\mathbb{R}^d)$ such that $\lVert \chi_{B_k}(f)-f_{\epsilon} \rVert \leq \epsilon$ over $B_k$ which is of finite measure. Would probably use the fact that $\mathbb{R}^d=\bigcup_{k=1}^{\infty}B_k$. But I'm not sure if this is a good idea.
Thank you for your time.
For $k\in \mathbb{N}$ define $f_k\colon\mathbb{R}^n\to \mathbb{R}$, $f_k\colon= \mathbb{1}_{B_k(0)}f$. Then $f_k\to f$ pointwise in $\mathbb{R}^n$ and $|f_k|\leq |f|\in L^p(\mathbb{R}^n)$. By dominated convergence theorem it is $$\lim_{k\to \infty} \int_{\mathbb{R}^n} f_k(x) dx = \int_{\mathbb{R}^n} \lim_{k\to \infty} f_k(x) dx = \int_{\mathbb{R}^n} f(x) dx.$$