= 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.
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