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I'm trying to show that the space of $T$-invariant probability measures is compact in the weak* topology ($T$ is some measurable transformation from a compact metric space to itself). I'm trying to use a functional analysis method as that's what I'm most comfortable with. So suppose $m_n$ is a sequence of $T$-invariant measures converging to some measure $m$. Is it correct that:

$$\int f dm = \lim_{n \to \infty} \int f dm_n = \lim_{n \to \infty} \int f(T) dm_n = \int f(T) dm$$ with $f$ any continuous function from $X$ to reals.

from this we can infer that $m$ is $T$-invariant and hence the space of $T$-invariant measures is a closed subset in a compact space hence compact?

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We just need to use the fact that sequentially closed imply closed. Here, it's the case because we have metrizability, thanks to Prokohrov metric. See for instance here or Dudley's book Real Analysis and Probability, chapter 11.

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