English
If f is a.e. positive, then applying density f followed by density f⁻¹ returns the original measure.
Русский
Если f > 0 почти везде, то применение плотности f затем плотности f⁻¹ возвращает исходную меру.
LaTeX
$$ If f is a.e. positive, then (μ.withDensity f).withDensity f⁻¹ ≤ μ and in fact equals μ under appropriate conditions. $$
Lean4
theorem withDensity_inv_same_le {μ : Measure α} {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) :
(μ.withDensity f).withDensity f⁻¹ ≤ μ :=
by
change (μ.withDensity f).withDensity (fun x ↦ (f x)⁻¹) ≤ μ
rw [← withDensity_mul₀ hf hf.inv]
suffices (f * fun x ↦ (f x)⁻¹) ≤ᵐ[μ] 1 by
refine (withDensity_mono this).trans ?_
rw [withDensity_one]
filter_upwards with x
simp