Left continuity of $L^p(E)$ norm when $p < 1$.

Question

I'm stuck on the final stages of a homework problem. Given $\mu(E) < \infty$ and $f$ measurable, we've been asked to show that

$\lim_{p \to p_0^-} \lVert f \rVert_p = \lVert f \rVert_{p_0}$.

Initially, I just approached this problem as a direct proof using Hölder's inequality, but that obviously doesn't work when $p_0 < 1$, so now I've changed tactics a bit.

I've handled the case where $f \notin L^{p_0}(E)$, I've shown that $f \in L^p$ if $f \in L^{p_0}$, and I've used DCT to arrive at the conclusion that $\lim_{p \to p_0^-} \int_E \lvert f \rvert^p = \int_E \lvert f \rvert^{p_0}$.

I'm running into trouble concluding from here that my limit is preserved by taking the $1/p$ and $1/p_0$ exponents on each side when $p_0 < 1$.

In the other questions I've looked at, either I couldn't figure out from the answers how the question of convergence was addressed in the case $p < 1$, or it wasn't a consideration to begin with.

Any help would be appreciated -- thank you!

Answer 1

Of course that the limit is preserved, note that the potentiation by a positive number is a continuous function in $[0,\infty)$, that is, the function $g:(0,\infty)\to(0,\infty),\, p\mapsto x^p$ is continuous for all $x\geqslant 0$ and so $$ \lim_{p\to p_0}\left(I_p\right)^{1/p}=\exp\left(\lim_{p\to p_0}\frac{\ln I_p}{p}\right)=\exp\left(\frac{\ln (\lim_{p\to p_0}I_{p})}{\lim_{p\to p_0}p}\right)=\exp\left(\frac{\ln I_{p_0}}{p_0}\right) $$ where $I_p:=\int |f|^p$.

Answer 2

Hint; write $\|f\|_p = \exp(\frac 1p \log\int_E |f|^p) $ and then use the continuity of the exponential.

Answer 3

Assuming that probability measure is in question and let $N$ be fixed, we let $\varphi(u)=u^{p_{0}/p}$, Jensen's inequality gives \begin{align*} \varphi\left(\int|f|^{p}\chi_{|f|\leq N}\right)\leq\int\varphi(|f|^{p}\chi_{|f|\leq N}), \end{align*} then \begin{align*} \left(\int|f|^{p}\chi_{|f|\leq N}\right)^{1/p}\leq\left(\int|f|^{p_{0}}\chi_{|f|\leq N}\right)^{1/p_{0}}\leq\left(\int|f|^{p_{0}}\right)^{1/p_{0}}, \end{align*} and hence \begin{align*} \limsup_{p\rightarrow p_{0}^{-}}\|f\|_{L^{p}}=\limsup_{p\rightarrow p_{0}^{-}}\lim_{N\rightarrow\infty}\|f\chi_{|f|\leq N}\|_{L^{p}}\leq\|f\|_{L^{p_{0}}}. \end{align*} On the other hand, let $N$ be fixed again and we have \begin{align*} \|f\chi_{|f|\leq N}\|_{L^{p_{0}}}&=\left(\int_{|f|\leq N}\lim_{p\rightarrow p_{0}^{-}}|f|^{p}\right)^{1/p_{0}}\\ &\leq\liminf_{p\rightarrow p_{0}^{-}}\left(\int_{|f|\leq N}|f|^{p}\right)^{1/p_{0}}\\ &=\liminf_{p\rightarrow p_{0}^{-}}\left(\left(\int_{|f|\leq N}|f|^{p}\right)^{1/p}\right)^{p/p_{0}}\\ &\leq\limsup_{p\rightarrow p_{0}^{-}}\left(\left(\int_{|f|\leq N}|f|^{p}\right)^{1/p}\right)^{p/p_{0}}\\ &\leq\limsup_{p\rightarrow p_{0}^{-}}\left(\left(\int_{|f|\leq N}|f|^{p_{0}}\right)^{1/p_{0}}\right)^{p/p_{0}}\\ &=\left(\int_{|f|\leq N}|f|^{p_{0}}\right)^{1/p_{0}}\\ &=\|f\chi_{|f|\leq N}\|_{L^{p_{0}}}. \end{align*} From this we can see that \begin{align*} \lim_{p\rightarrow p_{0}}\|f\chi_{|f|\leq N}\|_{L^{p}}^{p/p_{0}}=\|f\chi_{|f|\leq N}\|_{L^{p_{0}}}. \end{align*} And we have \begin{align*} \log\|f\chi_{|f|\leq N}\|_{L^{p_{0}}}&=\log\left(\lim_{p\rightarrow p_{0}^{-}}\|f\chi_{|f|\leq N}\|_{L^{p}}^{p/p_{0}}\right)\\ &=\lim_{p\rightarrow p_{0}^{-}}\log\|f\chi_{|f|\leq N}\|_{L^{p}}^{p/p_{0}}\\ &=\lim_{p\rightarrow p_{0}^{-}}\dfrac{p}{p_{0}}\log\|f\chi_{|f|\leq N}\|_{L^{p}}\\ &\leq\liminf_{p\rightarrow p_{0}^{-}}\log\|f\chi_{|f|\leq N}\|_{L^{p}}\\ &\leq\limsup_{p\rightarrow p_{0}^{-}}\log\|f\chi_{|f|\leq N}\|_{L^{p}}\\ &\leq\limsup_{p\rightarrow p_{0}^{-}}\log\|f\chi_{|f|\leq N}\|_{L^{p_{0}}}\\ &=\log\|f\chi_{|f|\leq N}\|_{L^{p_{0}}}, \end{align*} we see that \begin{align*} \lim_{p\rightarrow p_{0}^{-}}\log\|f\chi_{|f|\leq N}\|_{L^{p}}=\log\|f\chi_{|f|\leq N}\|_{L^{p_{0}}}, \end{align*} and hence \begin{align*} \lim_{p\rightarrow p_{0}^{-}}\|f\chi_{|f|\leq N}\|_{L^{p}}=\|f\chi_{|f|\leq N}\|_{L^{p_{0}}}. \end{align*} Finally, \begin{align*} \|f\|_{L^{p_{0}}}&=\lim_{N\rightarrow\infty}\|f\chi_{|f|\leq N}\|_{L^{p_{0}}}\\ &=\lim_{N\rightarrow\infty}\lim_{p\rightarrow p_{0}^{-}}\|f\chi_{|f|\leq N}\|_{L^{p}}\\ &=\lim_{N\rightarrow\infty}\liminf_{p\rightarrow p_{0}^{-}}\|f\chi_{|f|\leq N}\|_{L^{p}}\\ &\leq\liminf_{p\rightarrow p_{0}^{-}}\|f\|_{L^{p}}\\ &\leq\limsup_{p\rightarrow p_{0}^{-}}\|f\|_{L^{p}}\\ &\leq\|f\|_{L^{p_{0}}}. \end{align*}

Edit:

Since \begin{align*} \lim_{p\rightarrow p_{0}}\|f\chi_{|f|\leq N}\|_{L^{p}}^{p/p_{0}}=\|f\chi_{|f|\leq N}\|_{L^{p_{0}}}, \end{align*} then \begin{align*} \lim_{p\rightarrow p_{0}}\log\|f\chi_{|f|\leq N}\|_{L^{p}}^{p/p_{0}}=\log\left(\lim_{p\rightarrow p_{0}^{-}}|f\chi_{|f|\leq N}\|_{L^{p}}^{p/p_{0}}\right)=\log\|f\chi_{|f|\leq N}\|_{L^{p_{0}}}. \end{align*} And hence \begin{align*} \lim_{p\rightarrow p_{0}}\log\|f\chi_{|f|\leq N}\|_{L^{p}}=\lim_{p\rightarrow p_{0}^{-}}\dfrac{p_{0}}{p}\log\left(|f\chi_{|f|\leq N}\|_{L^{p}}^{p/p_{0}}\right)=\log\|f\chi_{|f|\leq N}\|_{L^{p_{0}}} \end{align*} and \begin{align*} \lim_{p\rightarrow p_{0}}\|f\chi_{|f|\leq N}\|_{L^{p}}&=\lim_{p\rightarrow p_{0}^{-}}\exp\log\left(|f\chi_{|f|\leq N}\|_{L^{p}}\right)\\ &=\exp\left(\lim_{p\rightarrow p_{0}^{-}}\log|f\chi_{|f|\leq N}\|_{L^{p}}\right)\\ &=\|f\chi_{|f|\leq N}\|_{L^{p_{0}}}. \end{align*} The truncation $\chi_{|f|\leq N}$ throughout is to make sure those integrals are of finite, or else using Jensen's inequality and the continuity of $\log$ or $\exp$ would be a little problematic.