Properties of the functions $f(x-t)$ for $t \in \mathbb{R}$

Solution 1:

One can easily extend the basic theorems MCT, Fatou, DCT to this situation. For example, $f_t\to f$ in $L^p$ as $t\to 0$ means that the function $\phi:\Bbb{R}\to\Bbb{R}$ defined as $\phi(t):=\|f_t-f\|_p$ is such that $\lim\limits_{t\to 0}\phi(t)=0$. For such real functions, checking something about limits is the same as checking along every sequence. Meaning that \begin{align} \lim_{t\to 0}\phi(t)&=0 \end{align} if and only if for every sequence $\{t_n\}_{n=1}^{\infty}$, with $\lim\limits_{n\to\infty}t_n=0$, we have \begin{align} \lim_{n\to\infty}\phi(t_n)&=0. \end{align} So, this allows you to reduce to the case of sequences and thus use your usual sequential variant of DCT/MCT/Fatou.