Measurable Hermitian-positive functions

Let $\mathfrak B^c_a$, $\mathfrak B_a^m$, $\mathfrak B_a^s$ ($0<a\le\infty$), respectively, denote the sets of continuous, measurable, and almost-everywhere vanishing functions $f(х)$ ($-a<x<a$; $f(0)>0$). The theorem is proved that for every $f\in\mathfrak B_a^m\setminus(\mathfrak B_a^c\cup\mathfrak B_a^s)$ there correspond $f_c\in\mathfrak B_a^c$ and $f_s\in\mathfrak B_a^s$, such that $f=f_c+f_s$ Some unsolved problems related to this theorem are formulated.