Can a convex combination of non-convex non-negative functions be convex function? It is well known that a convex combination of convex functions is still a convex functions. I am wondering is there possible that a convex function can be represented as a convex combination of both non-convex and convex and non-negative functions? Here, the concept of convex combination is relaxed to include an integration form, i.e. $\int_{a} f_a(x)d \mu(a)$ where $\mu$ is a probability measure satisfying  $\int_{a} d\mu(a) = 1$.
This is always possible. You can write the convex function $f$ as
$$
f = \frac23 (2f)  + \frac13(-f).
$$
Not always. Here is a counterexample: $f(x)=x^2$ (convex), $g(x)=-x^4$ (non-convex, concave) and the convex combination
$$
1/2\, f(x) + 1/2\, g(x)= \frac{x^2-x^4}2
$$
which is non-convex (and concave).