Theorems · Theorem · measure theory
MeasureTheory.tilted_apply_eq_ofReal_integral
∀ {α : Type u_1} {mα : MeasurableSpace α} {μ : MeasureTheory.Measure α} [MeasureTheory.SFinite μ] (f : α → ℝ)
(s : Set α), (μ.tilted f) s = ENNReal.ofReal (∫ (a : α) in s, Real.exp (f a) / ∫ (x : α), Real.exp (f x) ∂μ ∂μ)- Defined in
- Mathlib.MeasureTheory.Measure.Tilted
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 257 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- MeasureTheory.SFinite
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites27
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Setstatement and proof · cited by 53,352
- Realstatement and proof · cited by 25,697
- MeasurableSpacestatement and proof · cited by 13,106
- MeasureTheory.Measurestatement and proof · cited by 10,939
- ENNRealstatement and proof · cited by 9,879
- MeasureTheory.integralstatement and proof · cited by 1,779
- MeasureTheory.Measure.restrictstatement and proof · cited by 1,646
- MeasureTheory.Integrableproof · cited by 1,367
- le_of_ltproof · cited by 1,175
- Real.expstatement and proof · cited by 871
- ENNReal.ofRealstatement and proof · cited by 863
Cited by1
Results whose statement or proof uses this declaration.
- ProbabilityTheory.tilted_mul_apply_eq_ofReal_integral_mgfproof · cited by 1