Homework 7

Due Date: Friday, April 8

1. If \((X,\mu)\) is a measure space, a measure-preserving transformation of \(X\) is a bijection \(\varphi\colon X\to X\) such that \(\varphi^{-1}(E)\) is measurable for each measurable set \(E\subseteq X\), with \(\mu\bigl(\varphi^{-1}(E)\bigr) = \mu(E)\).

   Let \((X,\mu)\) be a measure space, and let \(\varphi\) be a measure-presrving transformation of \(X\). Prove that if \(f\) is a Lebesgue integrable function on \(X\), then \(f\circ \varphi\) is also Lebesgue integrable, and \[ \int_X (f\circ \varphi)\,d\mu \,=\, \int_X f\,d\mu. \]

2. 1. Prove that for any measurable set \(E \subseteq \mathbb{R}\), there exists a sequence \(\{g_n\}\) of continuous functions so that \[ \lim_{n\to\infty} \int_{\mathbb{R}} |g_n-\chi_E|\,dm \,=\, 0. \] (Hint: Look up Urysohn's lemma, which is Theorem 33.1 in Munkres.)
   2. Prove that for any \(L^1\) function \(f\colon \mathbb{R}\to\mathbb{R}\), there exists a sequence \(\{g_n\}\) of continuous functions so that \[ \lim_{n\to\infty} \int_{\mathbb{R}} |g_n-f|\,dm \,=\, 0. \]
3. Let \((X,\mu)\) be a measure space with \(0 < \mu(X) < \infty\), and let \(f\colon X\to[0,\infty)\) be a bounded measurable function. Define \[ \|f\|_p \,=\, \biggl(\int_X f^p\biggr)^{1/p} \] for each \(p \in [1,\infty)\), and let \[ \|f\|_{\infty} \,=\, \min\bigl\{M \;\bigr|\; f \leq M\text{ almost everywhere}\bigr\}. \]
   1. Prove that \(\|f\|_p \leq \|f\|_\infty\, \mu(X)^{1/p}\) for all \(p\geq 1\).
   2. Prove that \(\|f\|_p \to \|f\|_\infty\) as \(p\to\infty\).
      (Hint: Let \(\alpha\in (0,1)\) and prove that \(\|f\|_p \geq \alpha \|f\|_\infty\) for sufficiently large \(p\).)