I'm not sure how one can proof the following statement:
We have a probability space $(\Omega, \mathbb{F}, \mathbb{P})$ and a $\mathbb{F}$-measurable random variable $X$. Furthermore we have a set of measures equivalent to $\mathbb{P}$, denoted by $\mathcal{M}$, which is assumed to be non-empty and convex. The statement now reads as follows:
$\sup_{\mathbb{Q} \in \mathcal{M}} E^\mathbb{Q}[X] < \infty$ and there is some $\mathbb{Q}^* \in \mathcal{M}$ with $E^{\mathbb{Q}^*} [X] = \sup_{\mathbb{Q} \in \mathcal{M}} E^\mathbb{Q}[X]$ if and only if the map $\mathcal{M} \to \mathbb{R}, \mathbb{Q} \mapsto E^\mathbb{Q}[X]$ is finite and constant.
The implication $\Leftarrow$ is clear, but I don't know how the converse is proved.
Also, I'm not even sure if the statement is true for this setting, as the original problem is quite large (I want to do exercise 1.4.2. (d) in Föllmer Schied, Stochastic finance, an introduction in discrete time).
Does anyone have an idea on how this might be proved? Or is this not true in this setting? Thank you!