setIntegral_toReal_rnDeriv_eq_withDensity'

English

With Lebesgue decomposition, for a measurable set s, the real-valued integral of the RN-deriv equals theReal of density under ν.

Русский

При разложении Лебега, для измеримого множества s интеграл по RN-дериватива в rzeczyвном виде равен real-density под ν.

LaTeX

$$$\\int_{s} (μ.rnDeriv ν(x))^{\\toReal} dν(x) = μ.real(s)$$$

Lean4

theorem setIntegral_toReal_rnDeriv_eq_withDensity' [SigmaFinite μ] {s : Set α} (hs : MeasurableSet s) :
    ∫ x in s, (μ.rnDeriv ν x).toReal ∂ν = (ν.withDensity (μ.rnDeriv ν)).real s :=
  by
  rw [integral_toReal (Measure.measurable_rnDeriv _ _).aemeasurable, measureReal_def]
  · rw [ENNReal.toReal_eq_toReal_iff, ← withDensity_apply _ hs]
    simp
  · exact ae_restrict_of_ae (Measure.rnDeriv_lt_top _ _)

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