Relation between a convex lower semi-continuous function and its integral functional.

Question

I'm looking for a reference for the following theorem.

In some multidimensional calculus of variations script I found the following theorem.

Theorem: Let $\Omega \subset \mathbb{R}^n$ be open and bounded, $f : \Omega \rightarrow [0,\infty]$ Borel measurable, $p \in [0,\infty]$. Let $E$ be defined as $E : L^p(\Omega) \rightarrow \mathbb{R}$ s.t. $E(u) = \int_{\Omega} f(u(x))dx$.

Then following statements are equivalent:

1. $E$ is lower semi-continuous w.r.t. the weak* $L^p$ topology.
2. $f$ is lower semi-continuous and convex.

The script provides a semi-complete proof of this theorem, the direction "$1 \Rightarrow 2$ " is not too complicated, while the other one is a bit involved. Does someone have any reference to this result?

I'm especially interested in the implication "$ 2 \Rightarrow 1 $", for the $p = \infty $ case.

Many thanks in advance!

Answer 1

Ok I found the reference. It's called Tonelli's Theorem, see Wikipedia. A proof can be found here:

Renardy, Michael, and Robert C. Rogers. An introduction to partial differential equations. Vol. 13. Springer Science & Business Media, 2006 (Theorem 10.16, Page 347)

And it relies on Mazur's lemma.