Math 461 Bard College

# 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 $\int_X (f\circ \varphi)\,d\mu \,=\, \int_X f\,d\mu.$

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 $\lim_{n\to\infty} \int_{\mathbb{R}} |g_n-f|\,dm \,=\, 0.$
2. 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 $$p$$.)