For functions $(f_n)_{n \in \mathbb{N}} \subset L^p, f \in L^p$ prove equivalent condition to convergence in p-norm.

Question

For $p \in [1,\infty)$ let $f, f_n \in L^p(\mathbb{R})$ $\forall n \in \mathbb{N}$. Prove that TFAE:
$i) \: f_n \to f$ in $L^p(\mathbb{R})$ as $n \to \infty$
$ii) \: f_n \to f$ in $L^p([-N,N])$ as $n \to \infty$ $\forall N \in \mathbb{N}$ and $\lVert f_n \rVert_p \to \lVert f\rVert_p$.

I was able to show the implication $i) \implies ii)$.
For the other implication I tried:
$\int \limits_{\mathbb{R}} |f(x) -f_n(x)|^p dx = \int \limits_{\mathbb{R}} \lim \limits_{N \to \infty}\mathbf{1}_{[-N,N]}(x) \cdot |f(x) -f_n(x)|^p dx \\ = \lim \limits_{N \to \infty} \int \limits_{\mathbb{R}} \mathbf{1}_{[-N,N]}(x) |f(x) -f_n(x)|^p dx$
and then taking $n \to \infty$ exchanging the limits somehow.

The second equality is justified by dominated convergence, dominating $\mathbf{1}_{[-N,N]}(x) |f(x) -f_n(x)|^p$ by $|f(x) -f_n(x)|^p$ which is integrable.

Answer 1

This answer, as written at the time of acceptance, was incomplete. Thanks to @Ruy for the heads up.

For the other implication, use the following lemma:

Lemma: If $f \in L^p (\mathbb{R})$ and $E_N \uparrow \mathbb{R}$, then $$ \left( \int_{E_N^C} |f|^p \mathrm{d}\lambda \right)^{\frac{1}{p}} \to 0 $$ as $N \to \infty$.

To prove this lemma, define the measure $|f|^p \mathrm{d} \lambda$ and use the (upper) continuity of measures.

Now, define $E_N = [-N, N], g^N = f \cdot \chi_{E_N}$ and $g^N_n = f_n \cdot \chi_{E_N}$. By Minkowski's inequality, we have $$ ||f - f_n||_p \leq ||f - g^N||_p + ||g^N - g^N_n||_p + ||g^N_n - f_n||_p .$$

If we let $n \to \infty$, we know the central term goes to $0$. Our job now is to prove the rightmost term approaches $||f - g^N||_p$. After this is done, we will have the inequality $$ ||f - f_n||_p \leq 2 ||f - g^N||_p. $$ Then, letting $N \to \infty$ and using the mentioned lemma, the result is proven.

To prove that $||f_n - g_n^N||_p \to ||f - g^N||_p$ as $n \to \infty$, we note that $$ ||f_n||_p^p = \int_\mathbb{R} |f_n|^p \mathrm{d}\lambda = \int_{E_N} |f_n|^p \mathrm{d}\lambda + \int_{E_N^C} |f_n|^p \mathrm{d}\lambda = ||g_n^N||_p^p + ||f_n - g_n^N||_p^p, $$ So that we have, by both assumptions made in (ii): \begin{align} ||f_n - g_n^N||_p^p &= ||f_n||_p^p - ||g_n^N||_p^p \\ &\to ||f||_p^p - ||g^N||_p^p \\ &= \int_\mathbb{R} |f|^p \mathrm{d}\lambda - \int_{E_N} |f|^p \rm{d}\lambda \\ &= \int_{E_N^C} |f|^p \rm{d}\lambda \\ &= ||f - g^N||_p^p. \end{align}

Answer 2

Given $\varepsilon >0$, and observing that $$ \int_{\mathbb R}|f(x)|^pdx = \lim_{N\to \infty } \int_{[-N, N]}|f(x)|^pdx, $$ choose some $N\in {\mathbb N}$ such that $$ \Big|\int_{\mathbb R}|f(x)|^pdx -\int_{[-N, N]}|f(x)|^pdx\Big|< \varepsilon ^p. $$ Let $\chi $ be the characteristicstic function of $[-N, N]$, put $\tilde\chi = 1 - \chi $, and notice that the inequality above says that $\|\tilde\chi f\|<\varepsilon $.

Observe that for all $g$ in $L^p$ we have that $$ \|g\|^p = \|\chi g\|^p+ \|\tilde\chi g\|^p, \tag{$*$} $$ because $$ \|g\|^p = \int_{\mathbb R}|g(x)|^pdx = \int_{[-N, N]}|g(x)|^pdx + \int_{{\mathbb R}\setminus [-N, N]}|g(x)|^pdx = \|\chi g\|^p+ \|\tilde\chi g\|^p. $$

We next claim that $$ \lim_n \|\tilde\chi f_n\| < \varepsilon . \tag{$**$} $$ To prove it, plug $g=f_n$ in ($*$), leading up to $$ \lim_n \|\tilde\chi f_n\|^p = \lim_n \|f_n\|^p - \lim_n \|\chi f_n\|^p = \|f\|^p - \|\chi f\|^p = \|\tilde\chi f\|^p <\varepsilon , $$ where we have used that $\|f_n\|\to \|f\|$, and that $\chi f_n\to \chi f$ in $L^p$.

Next choose $n_0$ such that, for all $n\geq n_0$, on has that $\|\chi f-\chi f_n\|<\varepsilon $, and $\|\tilde\chi f_n\| < \varepsilon $. For all such $n$ we get $$ \|f-f_n\| $$$$ \leq \|f-\chi f\| + \|\chi f-\chi f_n\| + \|\chi f_n-f_n\| $$$$ = \|\tilde\chi f\| + \|\chi f-\chi f_n\| + \|\tilde\chi f_n\| < 3\varepsilon . $$ This concludes the proof.