Probability distribution on a functional space invariant to any permutation of the domain Let $X$ and $Y$ be two finite sets. Let $\mu$ be a probability distribution over $Y^X$.
Any permutation $\sigma$ on $X$ induces a mapping $f_\sigma: Y^X \to Y^X$ by $f \mapsto f\circ \sigma$.
Let $\mu \circ f_\sigma^{-1}$ be the probability measure on $Y^X$, induced by $f_\sigma$ and $\mu$. 
Is it true that $\mu \circ f_\sigma^{-1} = \mu$ for any permutation $\sigma$ on $X$, iff $\mu$ is the uniform distribution on $Y^X$?
The "if" part is obviously correct. I am not sure if the "only if" part is true. Thanks.

Hint: Consider a random variable $U\in E = Y^X$ with distribution $\mu$.
Write the image of the distribution 
$$
(p_e^U = P(U = e))_{e\in E}
$$
of $U$ under the action of $f_\sigma$. You shoud get a permutation of the initial family.
