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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