Lemmas about conditional probability

theorem ProbabilityTheory.cond_prod_univ {α : Type u_1} {β : Type u_2} {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {μ : MeasureTheory.Measure α} [MeasureTheory.SFinite μ] {ν : MeasureTheory.Measure β} [MeasureTheory.IsProbabilityMeasure ν] (s : Set α) : (μ.prod ν)[|s ×ˢ Set.univ] = μ[|s].prod ν

Conditioning a product measure on an event of the first coordinate amounts to conditioning the first measure.