Let $K$ is compact metric space. $B(K)$ is a $\sigma$-Borel algebra in $K$. Let $\mu$ is a positive measure on $K$. $\mu(K)=1$. Let $X$ is a Banach space.
Let: $G:K\times X\rightarrow \mathbb{R}, G(t,e)$. Assume:
$+ $ $G(.,e)$ is continuous $\forall e\in X$.
$+ $ $G(t,.)$ is linear and continuous $\forall t\in K$.
Prove that :
$$\sup_{e\in X,||e||\leq 1}\int_{K}G(t,e)d\mu(t)=\int_{K}\sup_{e\in X,||e||\leq 1}G(t,e)d\mu(t)=\int_{K}||G(t,.)||_{L(X,\mathbb{R})}d\mu(t)$$