lpMeasSubgroup — Mathlib

English

(A reiteration) The equivalence in ae_eq_trim_iff_of_aestronglyMeasurable holds as stated previously.

Русский

Повторение эквивалентности ae_eq_trim_iff_of_aestronglyMeasurable как ранее.

LaTeX

$$$\text{(повтор)}\; $$$

Lean4

/-- `lpMeasSubgroup F m p μ` is the subspace of `Lp F p μ` containing functions `f` verifying
`AEStronglyMeasurable[m] f μ`, i.e. functions which are `μ`-a.e. equal to
an `m`-strongly measurable function. -/
def lpMeasSubgroup (m : MeasurableSpace α) [MeasurableSpace α] (p : ℝ≥0∞) (μ : Measure α) : AddSubgroup (Lp F p μ)
    where
  carrier := {f : Lp F p μ | AEStronglyMeasurable[m] f μ}
  zero_mem' := ⟨(0 : α → F), @stronglyMeasurable_zero _ _ m _ _, Lp.coeFn_zero _ _ _⟩
  add_mem' {f g} hf hg := (hf.add hg).congr (Lp.coeFn_add f g).symm
  neg_mem' {f} hf := AEStronglyMeasurable.congr hf.neg (Lp.coeFn_neg f).symm

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