If $a_n\to 0$, does $f(x+a_n)-f(x)\to 0$ in the $L^1$ norm? I had this problem in my midterm exam the last week and still I can't handle it:
It is true that if $f \in L^{1}(\mathbb{R})$, (that is, $f: \mathbb{R} \rightarrow \mathbb{R}$ is Lebesgue measurable, $\int f  d \lambda < \infty$, where $\lambda$  is the Lebesgue measure), and  $a_{n} \subset \mathbb{R}$ is a sequence such that $a_n \rightarrow 0$, then $$ \lim_{n \rightarrow \infty} \int \mid f(x +a_n) -f(x) \mid d\lambda = 0 ?$$
This is trivially true if $f$ is continuous $\lambda$- almost everywhere.
For the general case I tried to use the Dominated Convergence Theorem over $f_n:= f(x +a_n) -f(x)$, but It seems false that $f_n \rightarrow 0$ $ \lambda$-a.e. even if $f = \chi_E$ is a characteristic function where E $\in \mathcal{L}$ is a Lebesgue measurable set. Please someone can give me some pointers on this problem? Thanks in advance!
My background: We are following Folland's Real Analysis Book, the last thing we saw was modes of convergence.
 A: Both the fact that $C_{c}(\mathbb{R})$ is dense in $L^{1}(\mathbb{R})$
and the proposition you are trying to prove rely on the regularity
of Lebesgue measure. To avoid possible circular reasoning, we try
to develop it from the ground. The following is only a sketch.
Step 1. Let $\mathcal{V}$ be the familiy of Lebesgue integrable functions
that satisfies your proposition. Prove that $\mathcal{V}$ is a $||\cdot||_{1}$-closed
vector subspace of $L^{1}$.
Step 2. Prove that $1_{(a,b)}\in\mathcal{V}$ for any $-\infty<a<b<\infty$.
Step 3. Prove that $1_{A}\in\mathcal{V}$ for any Lebesgue measurable
set $A$ with $\lambda(A)<\infty$. (It follows from outer-regularity:
$\lambda(A)=\inf\sum_{n=1}^{\infty}\lambda(I_{n})$, where the infimum
runs over all countable family of pairwisely disjoint open intervals
$\{I_{n}\mid n\in\mathbb{N}\}$ that satisfies $A\subseteq\cup_{n}I_{n}$.
In this step, note that $\sum_{n=N+1}^{\infty}\lambda(I_{n})$ is
small when $N$ is large. More explicitly, choose such covering
$\{I_{n}\mid n\in N\}$ for $A$ such that $\sum_{n}\lambda(I_{n})$
approximates $\lambda(A)$ well. Then further approximate $1_{A}\approx\sum_{n=1}^{N}1_{I_{n}}$
in $||\cdot||_{1}$-norm.)
Step 4. $f\in\mathcal{V}$ for any simple functions in $L^{1}$.
Step 5. $\mathcal{V}=L^1$
