  In this work, we study pmp actions of countable groups on arbitrary diffuse
probability spaces under the point of view of weak equivalence. We will show
that any such an action is weakly equivalent to an action on a standard
probability space. We also propose a metric on the space of actions modulo weak
equivalence which is equivalent to the topology of Ab\'{e}rt and Elek. We will
give a simpler proof of the compactness of the space, showing that convergence
is characterized by ultraproducts. Using this topology, we will show that a
profinite action is weakly equivalent to an ultraproduct of finite actions.
  Finally, combining our results with another result of Ab\'{e}rt and Elek, we
will obtain a corollary about sofic entropy. We will show that for free groups
and some property (T) groups, sofic entropy of profinite actions depends
crucially on the chosen sofic approximation.
