Let $f \in L^p(\mathbb{R})$ for some $1 \leq p < \infty$. Given $t \in \mathbb{R}$, we set $f_t(x) := f(x-t)$.
Here I'm not sure if we can view $\{f_t\}$ as a sequence of functions, since they are not enumerated in a discrete sense. Is it true that when $t \rightarrow 0$, we have $f_t \rightarrow f$ pointwise a.e. and $||f_t||_p \rightarrow ||f||_p$?
I'm also not sure if it's possible to apply results such as Dominated convergence theorem and Fatou's lemma when the functions are indexed like this, even if we were to set $g_t(x) := f(x-1/t)$ and let $t$ tend to infinity.
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.