Ir al contenido principal

## Due November 27

### Problem 1

If $f_n$ is a sequence of measurable functions, then $\{x:\lim f_n \text{ exists} \}$ is a measurable set.

### Problem 2

If $f:\R\to\R$ is monotone, then is Borel measurable.

### Problem 3

If $f_n\in L^+$ decreases pointwise to $f$, and $\int f_1 < \infty$, then $\int f = \lim \int f_n$.

### Problem 4

Let $f_n\in L^1$ such that $f_n\rightrightarrows f$.

1. If $\mu(X)<\infty$, then $f\in L^1$ and $\int f_n\to\int f$.

2. If $\mu(X)=\infty$, then the conclusions of (1.) might fail.

### Problem 5

1. If $1\le p<r\le \infty$, $L^p\cap L^r$ is a Banach space with norm $||f||=||f||_p + ||f||_r$.

2. If $1\le p<q<r\le\infty$, the inclusion map $L^p\cap L^r\to L^q$ is continuous.