Bartle The Elements of Integration exercise 6S

Question

Let $f_n$ $\in$ $L_p(X,\chi,\mu)$, $1 \leq p <+\infty$ and let $\beta_n$ be defined for $E \in \chi$ by $$\beta_n(E) = \left(\int_{E} |f_n|^p d\mu\right)^{1/p}$$ and suposse that $(f_n)$ is a Cauchy sequence in $L_p$. If $\epsilon >0$, then exists a $\delta(\epsilon)>0$ such that if $E \in \chi$ and $\mu(E)<\delta(\epsilon)$, then $\beta_n(E)<\epsilon$ for all $n \in \mathbb{N}$.

I'm trying to solve the problem above, I already know that the limit of $\beta_n(E)$ exists for every $E \in \chi$ since $(f_n)$ is a Cauchy sequence. My attempt to solve the problem is to define the set $$B=\{ E \in \chi ; \beta_n(E)<\epsilon \,\,\,\forall n \in \mathbb{N}\,\,\, and\,\,\, \mu(E)<+\infty \}.$$ The set $B$ is clearly not empty and if $\mu(E) = 0$ for all $E \in B$ the proposition is proved in that case. The other case consists of the existence of one $M \in B$ such that $0<\mu(M)$, here is the part that I'm stuck and don't know how to proceed to prove the existence of such $\delta$. Every hint in how to use the hypothesis that $(f_n)$ is a Cauchy sequence to solve the second case will be much appreciated, thank you.

Answer 1

Here are some ideas. Observe that $$ \left\lvert \beta_n(E)-\beta_m(E)\right\rvert=\left\lvert \left\lVert f_n\mathbf 1(E)\right\rVert_p-\left\lVert f_m\mathbf 1(E)\right\rVert_p\right\rvert \leqslant \left\lVert (f_n-f_n)\mathbf 1(E)\right\rVert_p\leqslant \left\lVert f_n-f_m\right\rVert_p $$ hence for each positive $\varepsilon$, one can find $n_0$ such that for all $n\geqslant n_0$ and all $E\in\chi$, $$ \beta_n(E)\lt\varepsilon/2+\beta_{n_0}(E). $$ The problem thus reduces to shows the existence of a $\delta(\varepsilon)$ such that for all $E\in\chi$ such that $\mu(E)\lt\delta(\varepsilon)$, $$ \max_{1\leqslant j\leqslant n_0}\beta_j(E)<\varepsilon/2. $$ It suffices to do it for a single function because if $\delta_j$ does the job for $f_j$, then take $\delta=\min_{1\leqslant j\leqslant n_0}\delta_j$.