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Let $(X,\mu)$ be a $G$-space, i.e. a measure space with a Borel quasi-invariant $G$-action. Say that $X$ is amenable (equivalently, that the action is amenable) if there is a $G$-fixed point in every affine space over $X$ with an $\alpha$-twisted action, where $\alpha$ is a corresponding cocycle.

If $X$ is an amenable $G$-space, it follows more or less from definition that for every compact metric space $Y$ there is an $G$-equivariant measurable map $\varphi : X \to M(Y)$, where $M(Y)$ is the collection of probability measures on $Y$.

My question is - is this property equivalent to the above definition of amenable action?

Let $(X,\mu)$ be a $G$-space, i.e. a measure space with a Borel quasi-invariant $G$-action. Say that $X$ is amenable (equivalently, that the action is amenable) if there is a $G$-fixed point in every affine space over $X$ with an $\alpha$-twisted action, where $\alpha$ is a corresponding cocycle.

If $X$ is an amenable $G$-space, it follows more or less from definition that for every compact metric space $Y$ there is an $G$-equivariant measurable map $\varphi : X \to M(Y)$, where $M(Y)$ is the collection of probability measures on $Y$.

My question is - is this property equivalent to the above definition of amenable action or strictly weaker?