**Summary**


I would like to pass from a sequence of probability measures whose "limit" satisfies a desired property to a new probability measure that satisfies this property.


**Details**


We work on a probability space $(\Omega,\mathcal{F},P)$. Let $C\subseteq L^{\infty}$ be convex, contain $-L^{\infty}\_+$, and be closed in the weak\*-topology on $L^{\infty}$.


Suppose there exists a sequence of probability measures $(Q^n)\_{n\in\mathbb{N}}$, each of which is equivalent to $P$, such that for each $n$,
\begin{equation}
0\leq \sup\_{X\in C} E\_{Q^n}[X] \leq \frac{1}{n} \enspace (\*).
\end{equation}


The goal is to construct a probability measure $Q$, that is equivalent to $P$, such that
\begin{equation\*}
\sup\_{X\in C} E\_{Q}[X] =0.
\end{equation\*}


**Question**


The *intuitive idea* is to take the limit as $n\rightarrow \infty$ in $(\*)$. How can this be formalized? I.e. how can the probability measures $(Q^n)\_{n\in\mathbb{N}}$ be used to construct a new probability measure $Q$ with the desired properties? Are there any papers with a similar construction?



Construction of a probability measure from a sequence of probability measures
$\newcommand\R{\mathbb R}$This is impossible to do in such generality.


E.g., suppose that $\Omega=\R$, $\mathcal{F}$ is the Borel $\sigma$-algebra over $\R$, $P$ is the standard normal distribution,
$$C=\{g\in L^\infty\colon g\le f\},$$
$f$ is the standard normal pdf, and $Q^n$ is the normal distribution with mean $n$ and variance $1$.


Then $Q^n$ is equivalent to $P$ for each $n$ and
\begin{equation}
0\le\sup\_{X\in C} E\_{Q^n}X= E\_{Q^n}f
=\int\_\R f(t)f(t-n)\,dt
=\frac1{\sqrt2}f\Big(\frac n{\sqrt2}\Big)\le\frac1n
\end{equation}
for each $n$.
However, for any probability measure $Q$ (even if not equivalent to $P$),
\begin{equation\*}
\sup\_{X\in C} E\_Q X=E\_Q f>0,
\end{equation\*}
since $f>0$.


