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

Question

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.

A.s. convergence is not true but convergence of $L^{p}$ norms is true and it is proved in Rudin's RCA. — Kavi Rama Murthy, Jan 07 '22 at 05:20
@KaviRamaMurthy So I suppose that whenever a result states that "Let ${f_n}$ be a sequence of functions...", they automatically exclude functions of the above scenario? — oleout, Jan 07 '22 at 05:28
What is $t$ approaching in your limit? Moreover, $||f_t||_p = ||f||_p $ so the $L^p$ limit is true. — Dionel Jaime, Jan 07 '22 at 05:47
@DionelJaime it approaches 0, I left it out by mistake, edited — oleout, Jan 07 '22 at 06:04
Then pointwise is not true. It's true if $f$ is continuous at $x$. But there are $L^p$ functions which are discontinuous a.e. — Dionel Jaime, Jan 07 '22 at 06:10
$t\to 0$ is really a sequence not a net but the notation is hiding that implication slightly. — Cameron L. Williams, Jan 07 '22 at 06:16

score 2 · Accepted Answer · answered Jan 07 '22 at 05:48

2

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.

answered Jan 07 '22 at 05:48

peek-a-boo

65,833

I know that we have $||f_t||_p = ||f||_p$ since $L^p$-norms are translation invariant after a bit of recalling, but how does this show that $\lim _{t \rightarrow 0} ||f_t - f||_p = 0$? – oleout Jan 07 '22 at 06:12
@KelvinLian my answer here addresses your second query regarding applicability of DCT/Fatou to non-sequences. Your first part is called continuity of translation in $L^p$, and has been discussed several times on this site. The idea is to first prove it for $f\in C_c(\Bbb{R})$, and then use density to prove for all $f\in L^p(\Bbb{R})$. See Continuity of $L^1$ functions with respect to translation for the details. There, the proof is given for $L^1$, but $L^p$ is very similar. – peek-a-boo Jan 07 '22 at 06:17
Pointwise a.e convergence is not true btw (as you've already been told in the comments above). All we can deduce is there is a sequence ${t_n}{n=1}^{\infty}$ such that $t_n\to 0$ and $f{t_n}\to f$ pointwise a.e as $n\to\infty$. This is just a general measure theory fact. Convergence in $L^p$ implies some subsequence converges pointwise a.e. – peek-a-boo Jan 07 '22 at 06:21

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

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