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Homework 4, Real Analysis

Due September 14

Problem 1

  1. If the measurable $latex f_n\searrow f\ge 0$ with $latex \int f_1 < \infty$, then $latex \int f_n \to \int f.$

  2. Explain the condition $latex \int f_1 < \infty$.

Problem 2

  1. There exists a positive continuous $latex f \in L^1(\R)$ such that $latex \limsup_{|x|\to\infty} f(x) = \infty.$

  2. If $latex f\in L^1(\R)$ is uniformly continuous, then $latex \lim_{|x|\to\infty}f(x) = 0.$

Problem 3

If $latex f\in L^1(\R)$ and $latex F(x) = \int_{-\infty}^x f$. Then F is uniformly continuous.

Problem 4

Let $latex f:D\to\R$ be uniformly continuous, with $latex D\subset\R$.

  1. If $latex x_0$ is a limit point of D, then f has limit at $latex x_0$.

  2. f has a continuous extension to $latex \bar D$, the closure of D.