3. $L^p$ Space

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real analysis   Real and Complex Analysis  

$\newcommand{\argmin}{\mathop{\mathrm{argmin}}\limits}$ $\newcommand{\argmax}{\mathop{\mathrm{argmax}}\limits}$

As we defined the Lebesgue integral and proved the basic properties, it is time to study the space of functions with finite integrals.

Inequalities

$(X, \mathfrak{M}, \mu)$: a measure space where $\mu(X) = 1.$
$\varphi: \mathbb{R} \to \mathbb{R}$ is convex on $(a,b).$
$f: X \to (a,b).$ Then, $$ \varphi\left(\int f d\mu\right) \le \int \varphi \circ f d\mu. $$


$f, g \ge 0,$ $\frac{1}{p}+\frac{1}{q}=1,$ $p,q> 1,$ then $$ \int fg d\mu \le \left(\int f^p d\mu\right)^{1/p} \left(\int g^q d\mu\right)^{1/q}. $$


Let $E \sub H$ be a non-empty, closed, convex set. Then there uniquely exists $x \in E$ that satisfies $$ \|x\| = \inf_{y \in E} \|y\|. $$

$f,g \ge 0,$ $p \ge 1,$ then \(\int (f+g)^p d\mu \le \int f^p d\mu + \int g^p d\mu.\) </div>

$L^p$ space

$1\le p<\infty$

$L^p(\mu)$ is complete in $\|\cdot\|_p.$

Given a Cauchy sequence $(f_n) \in L^p(\mu)$ and $\epsilon>0.$ There exists a subsequence $(f_{n_k})$ such that $$ \|f_{n_{k+1}} - f_{n_{k}}\|_p < \frac{1}{2^k},~ \forall k. $$ And let $$ g_n = \sum_{k=1}^n |f_{n_{k+1}} - f_{n_{k}}|, \\ g = \sum_{k=1}^\infty |f_{n_{k+1}} - f_{n_{k}}|. $$ Then clearly $g_n \uparrow g$ and $g_n \ge 0.$ By monotone convergence, $$ \|g_n\|_p \uparrow \|g\|_p \le 1 < \infty. $$ This implies $g < \infty$ almost everywhere. Now, note that $$ f_{n_{k}} = f_{n_1} + \sum_{i=1}^{k-1}\left( f_{n_{i+1}} - f_{n_{i}} \right) $$ and define $$ f = f_{n_1} + \sum_{i=1}^{\infty}\left( f_{n_{i+1}} - f_{n_{i}} \right) $$ Then $$ \begin{aligned} |f| \le |f_{n_1}| + g < \infty ~\text{ a.e.} && \text{($|f_{n_1}|, g < \infty$ almost everywhere)} \end{aligned} $$ Thus $f_{n_k} \to f$ almost everywhere as $k \to \infty.$ Our claim is that $f_n \overset{L^p}{\to} f.$
For $\epsilon > 0,$ there exists $N\in\mathbb{N}$ such that $\|f_n - f_{n_k}\|_p < \epsilon$ for all $n, k \ge N.$ $$ \begin{aligned} \int |f_n - f|^p d\mu &= \int \liminf_k |f_n - f_{n_k}|^p d\mu \\ &\le \liminf_k \int |f_n - f_{n_k}|^p d\mu \\ &< \epsilon^p ~~\text{for $n \ge N$}. \end{aligned} $$ $$ \therefore \|f_n - f\|_p < \epsilon,~ \forall n \ge N. $$


$p=\infty$

$L^\infty(\mu)$ is complete with respect to $\|\cdot\|_\infty.$

uniformly cauchy


Continuous approximation

$L^p(\mu)$ and $C_c(X)$

  • $S$ ~ $L^p$
  • Lusin’s theorem
  • $C_c(X)$ ~ $L^p$

$C_0(X)$ and $C_c(X)$

  • $C_c$ is dense in $C_0$
  • $C_0$ is complete


References

  • Rudin. 1986. Real and Complex Analysis. 3rd edition. McGraw-Hill.
  • Real Analysis (Spring, 2021) @ Seoul National University, Republic of Korea (instructor: Prof. Insuk Seo).