# Homework 11, Real Analysis 2

## Due May 11

### Problem 1

Let $\rho:\R^d\to\R^d$ be a rotation. Then it induces a measure-preserving map of the sphere $\mathbb S^{d-1}$ with its measure $\sigma$.

### Problem 2

Use the polar coordinate formula to prove the following statements.

1. $\displaystyle \int_{\R^2} e^{-\pi |x|^2} dx = 1$
2. $\displaystyle \int_{\R^d} e^{-\pi |x|^2} dx = 1$ for any d.
3. $\sigma(\mathbb S^{d-1}) = \dfrac{2\pi^{d/2}}{\Gamma(d/2)}$.
4. $\displaystyle m(\mathbb B^d) = \frac{\pi^{d/2}}{\Gamma(d/2+1)}.$

### Problem 3

If $\mu$ is a finite Borel measure on the interval $[a,b]$, then

$\displaystyle f\mapsto l(f) = \int_a^b f d\mu$

is a linear functional on $C([a,b])$, positive in the sense that $l(f)\ge 0$ if $f\ge 0$.

Conversely, if l is a positive linear functional on $C([a,b])$, there exists a unique finite Borel measure $\mu$ on $[a,b]$ such that $l(f) = \int f d\mu$ for every $f\in C([a,b])$.

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