Spring 2018 - Problem 5

Solution.

By Riesz representation, we see $\mu \in \p{C\p{\br{0,1}}}^*$, so it suffices to show that $\int_0^1 f\p{t} \,\diff\mu\p{t} = 0$ for all $f \in C\p{\br{0,1}}$.

First, observe that by the mean value theorem, if $\abs{h} < x - 1$, then there exists $\xi \in \br{1, x}$ so that

Since $\mu\p{\br{0,1}} \in \R$, in particular $\mu$ is finite, and so constants are in $L^1\p{\mu}$. By dominated convergence,

Indeed, this is true for $n = 1$ by assumption, and it follows for all $n \in \N$ by induction. Hence, we have shown that

for all $n \geq 1$. Now let $A$ be the set of all finite $\R$-linear combinations of $\frac{1}{\p{2 + t}^n}$ for $n \geq 1$. It's clear that $A$ is a subalgebra of $C\p{\br{0,1}}$, that it vanishes nowhere, and that it separates points (e.g., $\frac{1}{2 + t}$ does both). By Stone-Weierstrass, we see that $A$ is dense in $C\p{\br{0,1}}$. Since $\mu$ is $0$ on a dense subset of $C\p{\br{0,1}}$, it follows that $\mu$ is $0$ on all of $C\p{\br{0,1}}$, and so $\mu = 0$, which was what we wanted to show.