Show that $\{f_n \} \to f$ in $L^p(E)$ iF $\{f_n\}$ belongs to and is bounded as a subset of $L^{p+\theta} (E)$.

Question

Assume $E$ has finite measure and $1 \leq p < \infty$. Suppose $\{ f_n\}$ is a sequence of measurable functions that converges pointwise a.e. on $E$ to $f$. For $1 \leq p < \infty$, show that $\{f_n \} \to f$ in $L^p(E)$ if there is a $\theta > 0$ such that $\{f_n\}$ belongs to and is bounded as a subset of $L^{p+\theta} (E)$.}

My attempt:

We have to prove two things: $\{f_n\}$ and $f$ belongs to $ L^{p}$ and $f_n \to f$ in $L^p(E)$.

For the first part: consider $\theta>0$ exists and $f_n \in L^{p+\theta}(E)$ we know that if $1\leq p \leq p+ \theta< \infty$ and $\{f_n \} \in L^{p+\theta}(E)$ $\Rightarrow$ $\{f_n \} \in L^{p}(E)$

(I used for this the following: Corollary 3: Let $E$ be a measurable set of finite measure and $1 \leq p1 < p2 \leq \infty$. Then $L^{p_2}(E) \subseteq L^{p_1}(E)$. Furthermore, $||f||_{p_1} \leq c||f||_{p_2} $ for all $f \in L^{p_2}(E)$)

How do I prove that $f \in L^p(E)$ and $f_n \to f$ in $L^p(E)$

Answer 1

Hints: 1. To prove $f \in L^p,$ use Fatou's lemma. 2. For $f_n\to f$ in $L^p,$ use Egorov to see $f_n \to f$ uniformly on $E$ save for a smallish subset of $E.$

Answer 2

It's just a (hard-analytic) comment to zhw's answer. As he had shown, $f\in L^p$. We note that since $E$ has finite measure, the topology of convergence in measure is induced by $d_m(f,g)=\int_E\min(\lvert f-g\rvert,1)$. What we need to do is to interpolate $\lVert f\rVert_{L^q}$ between $\lVert f\rVert_{L^p}$ and $\lVert f\rVert_m:=\int_E\min(\lvert f\rvert,1)$ for $1\le q<p$. Write $f=f_0+f_\infty$, where $$f_0(x)=\begin{cases} f(x)&x\le1\\ 0&\text{otherwise} \end{cases}$$ $$f_\infty(x)=\begin{cases} 0&x\le1\\ f(x)&\text{otherwise} \end{cases}$$ Let $\alpha_0=\int_{\lvert f\rvert\le1}1$, $\alpha_\infty=\lVert f_\infty\rVert_{L^1}$, $\beta_0=\lVert f_0\rVert_{L^p}^p$, $\beta_\infty=\lVert f_\infty\rVert_{L^p}^p$. It follows from Hölder's inequality that $\lVert f_0\rVert_{L^q}^q\le\alpha_0^{1-q/p}\beta_0^{q/p}$ and $\lVert f_\infty\rVert_{L^q}^q\le\alpha_\infty^{(1-q/p)/(p-1)}\beta_\infty^{(q-1)/(p-1)}$ (the later is also known as interpolation of $L^p$ spaces. Hence we can get an estimation of $\lVert f\rVert_{L^q}$ by $\alpha_0,\alpha_\infty,\beta_0,\beta_\infty$. Since $\alpha_0,\alpha_\infty\le\lVert f\rVert_m$ and $\beta_0,\beta_\infty\le\lVert f\rVert_{L^p}^p$, we obtain what we want. Moreover, we can optimize the estimation by a wise choice of $\alpha_0,\alpha_\infty$ and $\beta_0,\beta_\infty$ subject to $\alpha_0+\alpha_\infty=\lVert f\rVert_m$ and $\beta_0+\beta_\infty=\lVert f\rVert_{L^p}^p$.