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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.