Lorena Beard

2022-07-06

Let $f\in {L}^{p}\left(\mathbb{R}\right)$ for some $1\le p<\mathrm{\infty }$. Given $t\in \mathbb{R}$, we set ${f}_{t}\left(x\right):=f\left(x-t\right)$.
Here I'm not sure if we can view $\left\{{f}_{t}\right\}$ as a sequence of functions, since they are not enumerated in a discrete sense. Is it true that when $t\to 0$, we have ${f}_{t}\to f$ pointwise a.e. and $||{f}_{t}|{|}_{p}\to ||f|{|}_{p}$?
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 ${g}_{t}\left(x\right):=f\left(x-1/t\right)$ and let $t$ tend to infinity.

vrtuljakc6

Expert

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 $\varphi :\mathbb{R}\to \mathbb{R}$ defined as $\varphi \left(t\right):=‖{f}_{t}-f{‖}_{p}$ is such that $\underset{t\to 0}{lim}\varphi \left(t\right)=0$. For such real functions, checking something about limits is the same as checking along every sequence. Meaning that
$\begin{array}{rl}\underset{t\to 0}{lim}\varphi \left(t\right)& =0\end{array}$
if and only if for every sequence $\left\{{t}_{n}{\right\}}_{n=1}^{\mathrm{\infty }}$, with $\underset{n\to \mathrm{\infty }}{lim}{t}_{n}=0$, we have
$\begin{array}{rl}\underset{n\to \mathrm{\infty }}{lim}\varphi \left({t}_{n}\right)& =0.\end{array}$
So, this allows you to reduce to the case of sequences and thus use your usual sequential variant of DCT/MCT/Fatou.

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