I am reading on how to calculate the cost of direct products in measurable group theory. In the proof by Gaboriau we have the following:
Let $\Gamma, \Delta$ be infinite countable groups. We want to prove that the cost of their direct product is one, i.e. $C(\Gamma \times \Delta)=1$. Note that if $E$ is generated by a free action of $\Gamma$ and $F$ by a free action of $\Delta$, then $E \times F$ is generated by a free action of $\Gamma \times \Delta$, so $C(\Gamma \times \Delta)=1$ follows from the proposition below.
Meanwhile, the proposition goes as follows:
Let $R, S$ be countable aperiodic Borel equivalence relations on $X, Y$ resp. and let $\mu$ be an $(R \times S)$-invariant finite measure on $X \times Y$. Then $C_{\mu}(R \times S)=\mu(X \times Y)$.
I don't quite see how the equivalence relation being induced by a free action is aperiodic. Why is the proposition applicable in the conditions of the theorem about the cost?
Just in case, an action is free, if the only element acting trivially is the identity. An equivalence relation is aperiodic if it contains only infinite equivalence classes.