English
In a standard setting, Finite Strongly Measurable is equivalent to Measurable via a Sigma-Finite requirement.
Русский
В стандартной постановке финитная сильная измеримость эквивалентна измеримости при условии сигма-ск finite.
LaTeX
$$$\text{FinStronglyMeasurable}(f,\mu) \iff \text{Measurable}(f)$ under Sigma-Finite μ$$
Lean4
theorem finStronglyMeasurable_iff_stronglyMeasurable_and_exists_set_sigmaFinite {α β} {f : α → β} [TopologicalSpace β]
[T2Space β] [Zero β] {_ : MeasurableSpace α} {μ : Measure α} :
FinStronglyMeasurable f μ ↔
StronglyMeasurable f ∧ ∃ t, MeasurableSet t ∧ (∀ x ∈ tᶜ, f x = 0) ∧ SigmaFinite (μ.restrict t) :=
⟨fun hf => ⟨hf.stronglyMeasurable, hf.exists_set_sigmaFinite⟩, fun hf =>
hf.1.finStronglyMeasurable_of_set_sigmaFinite hf.2.choose_spec.1 hf.2.choose_spec.2.1 hf.2.choose_spec.2.2⟩
