theorem
ProbabilityTheory.uniformOn_apply_singleton_of_mem
{Ω : Type u_1}
[MeasurableSpace Ω]
[MeasurableSingletonClass Ω]
{s : Set Ω}
{x : Ω}
(hx : x ∈ s)
(hs : s.Finite)
:
(ProbabilityTheory.uniformOn s) {x} = 1 / ↑(Nat.card ↑s)
theorem
ProbabilityTheory.uniformOn_apply_singleton_of_not_mem
{Ω : Type u_1}
[MeasurableSpace Ω]
[MeasurableSingletonClass Ω]
{s : Set Ω}
{x : Ω}
(hx : x ∉ s)
:
(ProbabilityTheory.uniformOn s) {x} = 0
theorem
ProbabilityTheory.uniformOn_apply_eq_zero
{Ω : Type u_1}
[MeasurableSpace Ω]
[MeasurableSingletonClass Ω]
{s t : Set Ω}
(hst : s ∩ t = ∅)
:
(ProbabilityTheory.uniformOn s) t = 0
theorem
ProbabilityTheory.uniformOn_of_infinite
{Ω : Type u_1}
[MeasurableSpace Ω]
[MeasurableSingletonClass Ω]
{s : Set Ω}
(hs : s.Infinite)
:
theorem
ProbabilityTheory.uniformOn_apply
{Ω : Type u_1}
[MeasurableSpace Ω]
[MeasurableSingletonClass Ω]
{s : Set Ω}
(hs : s.Finite)
(t : Set Ω)
:
instance
ProbabilityTheory.uniformOn.instIsProbabilityMeasure
{Ω : Type u_1}
[MeasurableSpace Ω]
[MeasurableSingletonClass Ω]
{s : Set Ω}
[Nonempty ↑s]
[Finite ↑s]
:
Equations
- ⋯ = ⋯
theorem
ProbabilityTheory.map_uniformOn_apply
{Ω : Type u_1}
{Ω' : Type u_2}
[MeasurableSpace Ω]
[MeasurableSingletonClass Ω]
[MeasurableSpace Ω']
[MeasurableSingletonClass Ω']
{s : Set Ω}
{f : Ω → Ω'}
(hmes : Measurable f)
(hf : Function.Injective f)
{t : Set Ω'}
(ht : MeasurableSet t)
:
(MeasureTheory.Measure.map f (ProbabilityTheory.uniformOn s)) t = (ProbabilityTheory.uniformOn (f '' s)) t
theorem
ProbabilityTheory.map_uniformOn
{Ω : Type u_1}
{Ω' : Type u_2}
[MeasurableSpace Ω]
[MeasurableSingletonClass Ω]
[MeasurableSpace Ω']
[MeasurableSingletonClass Ω']
{s : Set Ω}
{f : Ω → Ω'}
(hmes : Measurable f)
(hf : Function.Injective f)
: