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Homework 11, Real Analysis 2

Due May 11

Problem 1

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

Problem 2

Use the polar coordinate formula to prove the following statements.

  1. $latex \displaystyle \int_{\R^2} e^{-\pi |x|^2} dx = 1$

  2. $latex \displaystyle \int_{\R^d} e^{-\pi |x|^2} dx = 1$ for any d.

  3. $latex \sigma(\mathbb S^{d-1}) = \dfrac{2\pi^{d/2}}{\Gamma(d/2)}$.

  4. $latex \displaystyle m(\mathbb B^d) = \frac{\pi^{d/2}}{\Gamma(d/2+1)}.$

Problem 3

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

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

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

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