Convex , then also Measurable

Question

I was reading about Jensen's inequality and noticed that don't require $\phi$ to be measurable here: Wikipedia link.

Therefore, I guess that being convex implies being measurable somehow, but I have never actually seen this and this is why I wanted to ask here if this is true?

Answer 1

If $f$ is convex then $\{f<\alpha\}$ is either empty or an interval, hence measurable.

(It's an interval because it's a convex set: if $f(x)<\alpha$ and $f(y)<\alpha$ then $f((1-\lambda)x+\lambda y)\le (1-\lambda)f(x)+\lambda f(y)<(1-\lambda)\alpha+\lambda\alpha=\alpha$.)