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.