I am looking to prove that $\min_{a \in \mathbb{R}} E[f(a, X)] \geq E[\min_{a \in \mathbb{R}}f(a, X)]$ where $f: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$, $f$ is guaranteed to have a minimum, and $X$ is some random variable with an unknown distribution.
This looks similar to Jensen's Inequality for a concave function where $\min_{a \in \mathbb{R}(\cdot)}$ would be the function of interest. If this is the case, I am not sure how to prove that $min_{a \in \mathbb{R}}(\cdot)$ is concave. If there is a simpler way, then it is eluding me and any help is wanted.
In general, if $X \le Y$ and both have expectations you have $EX \le EY$.
Assuming the relevant expectations exist, for any $a'$ you have $E f(a',X) \ge E \min_a f(a,X)$ and so $\inf_{a'} E f(a',X) \ge E \min_a f(a,X)$.