I feel this question is very silly, but I want to make sure I understand all the definitions correct. We consider discrete time and a state space $S$ which we assume to be Polish, where $B$ denotes the Borel-sigma algebra of $S$.
So a Markov chain is stationary if $P(X_{t_1} \in A_1, X_{t_2} \in A_2, ... , X_{t_k} \in A_k) = P(X_{t_1+s} \in A_1, X_{t_2+s} \in A_2, ... , X_{t_k+s} \in A_k)$, where the $A_i \in B$, $t_i,s \in \mathbb{N}$. The initial distribution is $\mu(A) = P(X_0 \in A)$, so now obviously $P(X_t \in A) = P(X_0 \in A) = \mu(A)$? Since we have to show this as an exercise, I highly doubt that this is already the solution, but I don't see why.
Furthermore, by setting $t=1$ wouldn't that directly imply that $\mu$ is an invariant distribution? Also the transition kernels must be stationary.