Convergence in $L^p$ spaces for $0 < p \leq 1$.

Question

Question: Let $f_n$ and $\phi$ be measurable functions on $\mathbb{R}^d$ such that $\int_{\mathbb{R}^d} |f_n(x)|^p |\phi(x)|\, dx \leq M$ for all $n$ where $0 < p \leq 1$. Suppose $f_n \to f$ a.e. Show that $$ \int_{\mathbb{R}^d} [ |f_n(x)|^p - |f(x)|^p - |f_n(x)-f(x)|^p] |\phi(x)|\, dx \to 0$$ as $n \to \infty$.

My attempt: For any $n \in \mathbb{N}$, we have $$||f_n|^p - |f_n|^p|\leq |f - f_n|^p$$ and in particular $$ |f_n|^p - |f|^p - |f_n-f|^p = (|f_n|^p - |f|^p) + (- |f_n - f|^p) \leq 2 |f_n - f|^p$$ So it suffices to show that $\int_{\mathbb{R}^d} |f_n - f|^p |\phi|\to 0$ as $n \to \infty$. For each $k \in \mathbb{N}$ define $$ F_k := |\phi|(|f_k|^p + |f|^p), \quad F := |\phi|(2 |f|^p) $$ then $F_k \to F$ a.e. and by Fatou's lemma $\int_{\mathbb{R}^d} |f|^p |\phi| = \liminf_{k \to \infty} \int_{\mathbb{R}^d} |f_k|^p |\phi|\leq M < \infty$ so $|f|^p |\phi| \in L^1(\mathbb{R}^d)$ and in particular $F_k, F \in L^1(\mathbb{R}^d)$. Since $|f_n-f|^p |\phi| \leq (|f_n|^p + |f|^p)|\phi|$ for all $n$ and if $$\lim_{k \to \infty} \int_{\mathbb{R}^d} F_k = \int_{\mathbb{R}^d} F$$ then by generalised Lebesgue dominated convergence theorem we have $\int_{\mathbb{R}^d} |f_n - f|^p |\phi| \to 0$ as $n \to \infty$.

Therefore, is there any way I can prove the following limit holds $$\lim_{k \to \infty}\int_{\mathbb{R}^d} |\phi|(|f_k|^p + |f|^p)\, dx = \int_{\mathbb{R}^d} 2|\phi||f|^p\, dx$$

Or is there a simpler way to tackle this problem?

Answer 1

Consider the measure $\mu(dx):=|\phi(x)|\,dx$. The assumptions in the statement of the problem implies that $(f_n,f:n\in\mathbb{N})\subset L_1(\mu)$.

For $0<p\leq 1$, the function $\phi(t)=t^p$ is monotone increasing and subadditive on $[0,\infty)$, that is $$(a+b)^p\leq a^p+b^p,\qquad a,b\geq0.$$ Hence $$\big||f_n|^p -|f_n-f|^p \big|\leq |f|^p,$$ for $$|f_n|^p=|f_n-f+f|^p\leq(|f_n-f|+|f|)^p\leq |f_n-f|^p+|f|^p$$ and $$|f_n-f|^p\leq(|f_n|+|f|)^p\leq |f_n|^p+|f|^p.$$

Since $|f_n|^p-|f_n-f|^p\xrightarrow{n\rightarrow\infty}|f|^p$ a.s., and application of by dominated convergence yields $$\int_X(|f_n|^p -|f_n-f|^p)\,d\mu\xrightarrow{n\rightarrow\infty}\int|f|^p\,d\mu,$$ whence the statement in the problem follows.

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Your strategy works only if $\int|f_n|^p\,d\mu\xrightarrow{n\rightarrow\infty}\int|f|^p\,d\mu$ by Riesz' theorem (see also, Theorem 7, in Royden, H. L., and Ftizatrick, P. M., Real Analysis, 4th edition, Pearson, 2010, pp. 148. For example, for $p=1$, consider $\phi(x)=\mathbb{1}_{(0,1]}(x)$ and $f_n(x)=n\mathbb{1}_{(0,1/n]}$. Clearly $f_n\xrightarrow{n\rightarrow\infty}f=0$ and $\sup_n\int f_n\phi=1<\infty$. It is clear that $$\int\big(|f_n|-|f|-|f_n-f|\big)\phi\xrightarrow{n\rightarrow\infty}0$$ however $\int|f_n-f|\phi\not\rightarrow0$ as $n\rightarrow\infty$.