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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....

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No - the simplest example: $S=\{1,2,\dots\}$, $\mu_n=\frac1n \delta_n$, and $f(n)=n$.

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