Let $f(t)$ be a convex function and let $g(t)$ denote its running average. Here $g(t)$ can be shown to be $$g(t) = \int_{0}^{1}f(ts)ds$$ Here, $g(t)$ is to be proved to be convex. A nice answer to this question is already posted here. My question is can this statement be proved by the following steps : (1) A convex function determines an epigraph (2) The epigraph is a convex set. Under the above integral operator which is linear , convexity is preserved (3) We hence get another epigraph which determines the convex function $g(t)$
Are all my steps correct and can convexity of $g(t)$ be proved in this way ?