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I have a metric space with a probability measure (implying that the measure is regular). What conditions should I ask of the measure or of the metric to get that a function in $\mathcal{L}^2(d\mu)$ can be approximated in $\mathcal{L}^2$ by a uniformly continuous function?

Thanks

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    Can you define precisely what you mean by "regular"? Conventions vary.2011-02-04

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Hint: Suppose $f$ is orthogonal to all bounded, uniformly continuous functions. Show that $\int_F f(x)\mu(dx)=0$ for all closed sets $F$. Deduce that $\int_B f(x)\mu(dx)=0$ for all Borel sets $B$.

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If I understood it well (did I?), Byron Schmuland's solution needs no additional conditions on your space $X$, since it relies on the theorem of characterization of maximal orthonormal systems (of arbitrary cardinal), although you'd have to exhibit one such system inside the bounded and uniformly continuous functions.

An alternate solution could use Lusin's theorem, for which you need to throw in local compactness of $X$. For the relevant setup, you can follow Rudin's "Real and complex analysis", theorems 2.24 and 3.14 in the third edition. The advantage here is that you get your result cheaply (local compactness is not that weird for a base space ;-) and for any $1 \leq p < \infty$, not just for the Hilbert case.