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Problem statement:

Let $x_j$ be a sequence in $[0,1]$. Consider $$\lim_{N\to\infty}\frac{1}{N}\sum_{j=1}^N f(x_j)\hspace 2cm (\star).$$ If ($\star$) exists for all $f\in C[0,1]$ then there is a positive Borel measure $\mu$ on [0,1] with $\mu([0,1])=1$ such that ($\star)=\int_{[0,1]} f d\mu$ for all $f\in C[0,1]$. Furthermore, if $x_j\to x$ then ($\star$) exists for all $f$, and the above applies. Which $\mu$ do we get?

Note: There are other similar questions like this out there, but they don't ask you to show that $\mu([0,1])=1$ and/or ask about a subsequence instead of just a convergent subsequence.

Attempt:

Given that ($\star$) holds for all $f$ I can show that the the map that sends $f$ to ($\star$) is a positive linear functional. Applying (one version) of the Riez Representation Theorem gives a measure $\mu$ as needed. However, how do I know $\mu([0,1])=1$?

As for what happens when $x_j\to x$, I suspect since $f(x_j)\to f(x)$ and we are averaging that $(\star)=f(x)$.

ttt
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1 Answers1

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As for $\mu([0,1])=1$, consider the constant function $f(x) = 1$ on $x\in [0,1]$.

For the second statement, if $f\in C[0,1]$ then $f(x_j) \rightarrow f(x)$.

In this case, the limit will converge to $f(x)$.

Thus, the measure is the Dirac measure at the point $x$.

Sungjin Kim
  • 20,102
  • Constant function that makes sense. As for the other part, intuition says that knowing $f(x_j)\to f(x)$ and averaging gives $f(x)$ in the limit, but how to show this formally? – ttt Aug 05 '15 at 07:27
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    http://math.stackexchange.com/questions/565288/can-you-please-check-my-cesaro-means-proof – Sungjin Kim Aug 05 '15 at 07:30