Invariant tests factor through a maximal invariant

If $T$ is a maximal invariant for a group action, an invariant test function is constant on each group orbit and therefore equals $b\circ T$ for a function $b$ on the orbit space. Conversely every such function is invariant. Appropriate quotient measurability is understood. If the action is transitive on a null family of probability distributions, invariance also gives the same null law of $T$ under every member. Thus tests based on $T$ eliminate that null nuisance variation.