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I would like to show that for a given function $f\in B[a,b]$ there exists a sequence $\{f_n\}_{n\in\mathbb{N}}\subset C_c[a,b]$ such that $$ f_n\to f \ a.e.\ \text{as }n\to\infty\quad \& \quad\sup_n||f_n||_\infty<\infty, $$ where $B[a,b]$ is the space of real-valued bounded Borel measurable functions on the interval $[a,b]$ and $C_c[a,b]$ is the space of real-valued compactly supported continuous functions on $[a,b]$.

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  • $\begingroup$ Do you know convolution? $\endgroup$ – Del Jan 20 '17 at 16:32
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Hint:

Note that bounded Borel measurable functions on finite interval $[a,b]$ are Lebesgue integrable, i.e, $B[a,b]\subset L[a,b]$. And we know that funtions in $C_c[a,b]$ are dense in $L[a,b]$. Hence there exists a sequence of functions $\{f_n\}$ in $C_c[a,b]$, such that $$ \lim_{n\to\infty} f_n(x)=f(x),\ a.e.\ x\in [a,b] $$

To show that $$ \sup_n ||f_n||_\infty<\infty, $$ you must require that $$\sup_n ||f||_\infty<\infty.$$ Can you see why?

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