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 \[ \lim_{n\to\infty} \int_X f_n\,d\mu \;=\; \int_X f\,d\mu. \]
3. 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 \[ \lim_{n\to\infty} \int_0^n \Bigl(1+\frac{x}{n}\Bigr)^n e^{-2x}\,dx \;=\; 1. \]
4. 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 \[ \sum_{n=1}^\infty\, \int_X f_n\,d\mu \;=\; \int_X\, \sum_{n=1}^\infty f_n\,d\mu. \]