[Merged by Bors] - chore(measure_theory/function/conditional_expectation): change the definition of condexp and its notation #14010
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…finition of condexp and its notation (#14010) Before this PR, the conditional expectation `condexp` was defined using an argument `(hm : m ≤ m0)`. This changes the definition to take only `m`, and assigns the default value 0 if we don't have `m ≤ m0`. The notation for `condexp m μ f` is simplified to `μ[f|m]`. The change makes the proofs of the condexp API longer, but no change is needed to lemmas outside of that file. See the file `martingale.lean`: the notation is now simpler, but otherwise little else changes besides removing the now unused argument `[sigma_finite_filtration μ ℱ]` from many lemmas. Also add an instance `is_finite_measure.sigma_finite_filtration`: we had a lemma with both `is_finite_measure` and `sigma_finite_filtration` arguments, but the first one implies the other.
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Before this PR, the conditional expectation
condexpwas defined using an argument(hm : m ≤ m0).This changes the definition to take only
m, and assigns the default value 0 if we don't havem ≤ m0.The notation for
condexp m μ fis simplified toμ[f|m].The change makes the proofs of the condexp API longer, but no change is needed to lemmas outside of that file. See the file
martingale.lean: the notation is now simpler, but otherwise little else changes besides removing the now unused argument[sigma_finite_filtration μ ℱ]from many lemmas.Also add an instance
is_finite_measure.sigma_finite_filtration: we had a lemma with bothis_finite_measureandsigma_finite_filtrationarguments, but the first one implies the other.