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Let $C_0(R)$ be the set of all functions that are continuous on R and the values are zero except on a compact set. Is $C_0(R)$ a subset of $\tilde\varphi$ I need to prove it or counterexample.

N0te to self: $\tilde\varphi$ is the set of real valued functions g defined on $ R^n$ for which there is a non-decreasing sequence $(g_k)$ of step functions such that $\lim g_k(x)=g(x)$ for a.a.x and the sequence $\bigl(\int g_k\bigr)$ is bounded. I need help. Thank you.

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up vote 1 down vote accepted

Here's a solution much nicer than what was here previously:

Let's assume, for convenience of notation, that $f$ is compactly supported and continuous on $[0,1]$. For each $n$, let $I_{k,n} = [\frac{k}{2^n},\frac{k+1}{2^n}]$ and let $$g_n(x) = \sum_{i=0}^{2^n-1} \chi_{I_{k,n}} \sup_{I_{k,n}} f.$$ These are then step functions decreasing to $f$ for every $x$ but maybe the dyadics, but these are countable and hence have measure zero. Each $g_n$ obviously has $\int g_n < \infty$.

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Thank you very much! – Klara May 4 '13 at 1:09

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