Consider the space of sub-distributions (i.e. positive measures of variation norm lower than 1) over a discrete subset $S$ of $\mathbb{R}$ (this set is measured by its powerset).


Let $(\mu\_n)\_n$ be a sequence of sub-distributions converging to the sub-distribution $\mu$ under the TV norm.


Let take a function $f : S \to \mathbb{R}$ and assume that for all $m \in \mathbb{N}$:
$$
\sum\_n f(n) \mu\_m(f^{-1}(n)) = t 
$$


(it is generalisation of the expected value definition for sub-distributions)


Do we necessarily have $ \sum\_n f(n) \mu(f^{-1}(n)) = t $ ? (as a result of a limit exchange theorem)? 


If not, what kind of hypothesis would ensure this fact? Trivially having $f$ bounded is sufficient but it seems to be a very strong condition.... 



Convergence of generalized expectation under total variation norm
No - the simplest example: $S=\{1,2,\dots\}$, $\mu\_n=\frac1n \delta\_n$, and $f(n)=n$.


