Why is $\alpha(x) = P^{\alpha}(X_0 = x)$?

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Assume that $(X_n)$ is a sequence of random variables that are stationary with respect to $P^{\alpha}$, with $\alpha > 0$ being a stationary distribution. Then, why are we allowed to write $\alpha(x) = P^{\alpha}(X_0 = x)$?

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Since $(X_n)$ is stationary, $(X_n)$ are identically distributed. That is, for every $i,j$, $P^{\alpha}(X_i=x)=P^{\alpha}(X_j=x)$. Conclude that

\begin{align*} P^{\alpha}(X_0=x) = \lim_{n \rightarrow \infty}P^{\alpha}(X_n=x)=\alpha(x) \end{align*}