# a.e. convergence of uniformly bounded continuous functions to a bounded measurable function

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 $[a,b]$.

• Do you know convolution? – Del Jan 20 '17 at 16:32

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?