# Space of $T$-invariant probability measures is compact.

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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