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I have a convergence question in measure theory that requires assistance:

Let $1\leq p<\infty$. Suppose $f,\ f_n \in L^P$, and $f_n\to f$ in $L^P$. (i.e $(\int|f_n-f|^pd\mu)^{1\over p}\to 0$ as $n\to\infty$) Show that $\int|f_n|^pd\mu \to \int|f|^pd\mu$.

For $p=1$, $$\lvert\int|f_n|d\mu-\int|f|d\mu\rvert=\lvert\int(|f_n|-|f|)d\mu\rvert\leq\int\lvert|f_n|-|f|\rvert d\mu\leq\int|f_n-f|d\mu\to 0$$ hence obtaining the required inequality.

However, I have trouble with $p>1$.

Kindly advise. Thank you.

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Let us show it in two steps: $$ f_n\to f\;\;\text{in } L^p\;\;\Longrightarrow\;\;|f_n|^p\to |f|^p \;\;\text{in }L^1\;\;\Longrightarrow \;\;\int |f_n|^p\,\mathrm d\mu\to\int |f|^p\,\mathrm d\mu. $$

The last implication should be pretty straightforward. For the first implication we might use that for $x,y\in\mathbb{R}$ one has $$ ||x|^p-|y|^p|\leq p(|x|^{p-1}+|y|^{p-1})|x-y| $$ and hence $$ \begin{align} \int ||f_n|^p-|f|^p|\,\mathrm d\mu & \leq \int p(|f_n|^{p-1}+|f|^{p-1})|f_n-f|\,\mathrm d\mu \\ &=\int p|f_n|^{p-1}|f_n-f|\,\mathrm d\mu+\int p|f|^{p-1}|f_n-f|\,\mathrm d\mu\\ \end{align} $$ which by the use of Hölder's inequality is less than or equal to $$ p\left(\int |f_n|^{q(p-1)}\,\mathrm d\mu\right)^{1/q}\left(\int |f_n-f|^p\right)^{1/p}+p\left(\int |f|^{q(p-1)}\,\mathrm d\mu\right)^{1/q}\left(\int |f_n-f|\,\mathrm d\mu\right)^{1/p}\\ $$ where $q>1$ is chosen such that $\frac1p +\frac1q=1$. This simplifies to (recall that $q(p-1)=p$) $$ p\left[\left(\int |f_n|^p\,\mathrm d\mu\right)^{1/q}+\left(\int |f|^p\,\mathrm d\mu\right)^{1/q}\right]\left(\int |f_n-f|^p\,\mathrm d\mu\right)^{1/p}. $$ The last factor tends to $0$ as $n\to\infty$ by assumption, and thus we only need to argue that $$ \sup_n\int |f_n|^p\,\mathrm d\mu<\infty. $$ This i'll leave for you to prove.

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  • $\begingroup$ Hi. Thank you for the prompt reply. I suppose for the last part where you mention regarding the finiteness of the sup can be deduced from the assumption that fn's are in Lp? If the sup is not finite, that shows that the integrals of some fn's raised to the power of p are unbounded, which contradicts the assumption? $\endgroup$
    – Novice
    Nov 20, 2013 at 17:59
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    $\begingroup$ It follows from the fact that $\int |f_n|^p\,\mathrm d\mu\leq \left[\left(\int |f-f_n|^p\,\mathrm d\mu\right)^{1/p}+\left(\int |f|^p\,\mathrm d\mu\right)^{1/p}\right]^p$ by Minkowski's inequality, where the first term goes to $0$ and second term is finite. $\endgroup$ Nov 20, 2013 at 18:06
  • $\begingroup$ Cool. Thank you for your help.:) $\endgroup$
    – Novice
    Nov 20, 2013 at 18:12

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