Math 461 Bard College

# Homework 6

Due Date: Friday, March 18

1. Let $$(X,\mathcal{M},\mu)$$ be a measure space, let $$f$$ be a measurable function on $$X$$, and suppose that $$\int_E f\,d\mu = 0$$ for every measurable set $$E\subseteq X$$. Prove that $$f=0$$ almost everywhere.
2. Let $$(X,\mathcal{M},\mu)$$ be a measure space with $$\mu(X) < \infty$$. Let $$\{f_n\}$$ be a sequence of measurable functions on $$X$$, and suppose that $$\{f_n\}$$ converges uniformly to an $$L^1$$ function $$f$$. Prove that $\lim_{n\to\infty} \int_X f_n\,d\mu \;=\; \int_X f\,d\mu.$
1. Prove that $$\log(1+x) \leq x$$ for all $$x\in[0,\infty)$$.
2. Prove that if $$f\colon \mathbb{R} \to[0,\infty)$$ is a continuous function, then $\int_{[0,\infty)} f\,dm \;=\; \int_0^\infty \!f(x)\,dx,$ where the integral on the right is an improper Riemann integral.
3. Prove that $\lim_{n\to\infty} \int_0^n \Bigl(1+\frac{x}{n}\Bigr)^n e^{-2x}\,dx \;=\; 1.$
3. Let $$(X,\mathcal{M},\mu)$$ be a measure space, and let $$\{f_n\}$$ be a sequence of measurable functions on $$X$$. Suppose that the sum $$\sum_{n=1}^\infty f_n$$ converges pointwise, and that $\sum_{n=1}^\infty\, \int_X |f_n|\,d\mu \;<\; \infty.$ Prove that $\sum_{n=1}^\infty\, \int_X f_n\,d\mu \;=\; \int_X\, \sum_{n=1}^\infty f_n\,d\mu.$