trim_measurableSet_eq — Mathlib

English

Similarly to the previous left-tilted rnDeriv results, the real-valued version holds: (μ.tilted f).rnDeriv ν x is real-valued a.e. with expression exp(f(x))/(∫ exp(f) dμ) times (μ.rnDeriv ν x).toReal

Русский

Аналогично: вещественный RN-дериватив наклонённой меры слева имеет вид exp(f(x))/(∫ exp(f) dμ)·(μ.rnDeriv_ν(x)).toReal.

LaTeX

$$$\\big((μ^{\\ Tilted f})\\mathrm rnDeriv_{ν} (x)\\big)^{\\toReal} =_{ν}^{\\mathrm{ae}} \\dfrac{e^{f(x)}}{\\int e^{f} dμ} \\cdot (μ\\mathrm rnDeriv_{ν}(x))^{\\toReal}.$$$

Lean4

theorem trim_measurableSet_eq (hm : m ≤ m0) (hs : @MeasurableSet α m s) : μ.trim hm s = μ s := by
  rw [Measure.trim, toMeasure_apply (ms := m) _ _ hs, Measure.coe_toOuterMeasure]

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