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## Due date: November 24

### 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

Let $f\in L^1(\R)$ and $f^\delta(x) = f(\delta x)$. Then

$\displaystyle \int f^\delta = \frac{1}{\delta}\int f.$

### Problem 3

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 4

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