Homework 4, Real Analysis

Due September 14

Problem 1

  1. If the measurable f_n\searrow f\ge 0 with \int f_1 < \infty, then \int f_n \to \int f.
  2. Explain the condition \int f_1 < \infty.

Problem 2

  1. There exists a positive continuous f \in L^1(\R) such that \limsup_{|x|\to\infty} f(x) = \infty.
  2. If f\in L^1(\R) is uniformly continuous, then \lim_{|x|\to\infty}f(x) = 0.

Problem 3

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

Problem 4

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

  1. If x_0 is a limit point of D, then f has limit at x_0.
  2. f has a continuous extension to \bar D, the closure of D.
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