Notes on Chapter 3 (\(L^p\)-spaces) of Walter Rudin's Real and Complex Analysis.

Chapter 3

This is a short chapter, and begins by discussing convex functions, and some generalizations of inequalities:

Jensen's Inequality: Let \(\mu\) be a positive measure on a \(\sigma\)-algebra \(\mathfrak{M}\) on a set \(\Omega\), so that \(\mu(\Omega)=1\). If \(f\) is a real function in \(L^1(\mu)\), if \(\forall x\in\Omega,\ a<f(x)<b\), and if \(\varphi\) is convex on \((a,b)\) then

$$\varphi\left(\int_\Omega f\ d\mu\right)\leq\int_\Omega (\varphi\circ f)\ d\mu.$$ □

Hölder, Minkowski Inequalities: Let \(p,\ q\) be conjugate exponents (\(1/p+1/q=1\)), \(1<p<\infty\). Let \(X\) be a measure space, with measure \(\mu\). Let \(f\) and \(g\) be measurable functions on \(X\), with range in \([0,\infty]\). Then

$$\int_X fg\ d\mu\leq\left[\int_X f^p\ d\mu\right]^{1/p}\left[\int_X g^q\ d\mu\right]^{1/q}\qquad\text{\small(Hölder)}$$

$$\left[\int_X (f+g)^p\ d\mu\right]^{1/p}\leq\left[\int_X f^p\ d\mu\right]^{1/p}+\left[\int_X g^p\ d\mu\right]^{1/p}\qquad\text{\small(Minkowski)}$$ □

The \(L^p(\mu)\) norm and spaces are defined as usual (as equivalence classes of functions equal a.e., though tacitly this distinction is glossed over), and it can be shown it is a metric space. Additionally, \(\|fg\|_1\leq\|f\|_p\|g\|_q\) for \(1\leq p\leq\infty\), with \(p\) and \(q\) conjugate exponents, and \(f\in L^p(\mu)\) and \(g\in L^q(\mu)\).

It is then shown that \(L^p(\mu)\) is a complete metric space for all positive measures and \(1\leq p\leq\infty\). Furthermore, the complex measurable simple functions \(s\) on \(X\) such that \(\mu(\{x: s(x)\neq 0\})\leq\infty\) are dense in \(L^p(\mu)\ (1\leq p<\infty)\), which is a nice approximation result.

If instead of considering \(L^p(\mu)\) on an arbitrary measure space, we fix a locally compact Hausdorff space \(X\) and consider a measure \(\mu\) obtained from the Riesz representation theorem. Then we have a nice approximation:

Approximation by \(C_c(X)\): For \(1\leq p<\infty\), \(C_c(X)\) is dense in \(L^p(\mu)\). □

Since \(L^p(\mu)\) is complete here, it is the completion of \(C_c(X)\) under the \(L^p(\mu)\) metric.

Taking \(p=1\) and \(\mu\) to be the Lebesgue measure on \(\mathbb{R}\), this shows that the Lebesgue integrable functions are the completion of the metric space under the metric

$$\int_{-\infty}^\infty |f(t)-g(t)|\ dt$$

after identifying functions that are equal a.e.. In this way the Lebesgue integral is the 'correct' generalization of the Riemann integral.

In the case of \(p=\infty\), we have a slightly different situation. A complex function \(f\) on a locally compact Hausdorff space is said to vanish at infinity if for every \(\varepsilon>0\) there exists a compact \(K\) such that \(|f(x)|<\varepsilon\) for \(x\notin K\). Denote the class of continuous \(f\) on \(X\) that vanish at infinity by \(C_0(X)\), which in particular contains those with compact support. We then have the following:

Sup norm completion: If \(X\) is a locally compact Hausdorff space, then \(C_0(X)\) is the completion of \(C_c(X)\) relative to the metric defined by the supremum norm. □

Note that the essential supremum (\(L^\infty\)) coincides with the supremum on \(C_c(R_k)\).