File: Mathlib/MeasureTheory/Constructions/EventuallyMeasurable.lean

full_name: EventuallyMeasurableSpace
code: /-- The `MeasurableSpace` of sets which are measurable with respect to a given σ-algebra `m`
on `α`, modulo a given σ-filter `l` on `α`. -/
def EventuallyMeasurableSpace : MeasurableSpace α 
goal: MeasurableSpace α

full_name: EventuallyMeasurableSet
code: /-- We say a set `s` is an `EventuallyMeasurableSet` with respect to a given
σ-algebra `m` and σ-filter `l` if it differs from a set in `m` by a set in
the dual ideal of `l`. -/
def EventuallyMeasurableSet (s : Set α) : Prop 
goal: Prop

full_name: MeasurableSet.eventuallyMeasurableSet
code: theorem MeasurableSet.eventuallyMeasurableSet (hs : MeasurableSet s) :
    EventuallyMeasurableSet m l s 
goal: EventuallyMeasurableSet m l s

full_name: EventuallyMeasurableSpace.measurable_le
code: theorem EventuallyMeasurableSpace.measurable_le : m ≤ EventuallyMeasurableSpace m l 
goal: m ≤ EventuallyMeasurableSpace m l

full_name: eventuallyMeasurableSet_of_mem_filter
code: theorem eventuallyMeasurableSet_of_mem_filter (hs : s ∈ l) : EventuallyMeasurableSet m l s 
goal: EventuallyMeasurableSet m l s

full_name: EventuallyMeasurableSet.congr
code: /-- A set which is `EventuallyEq` to an `EventuallyMeasurableSet`
is an `EventuallyMeasurableSet`. -/
theorem EventuallyMeasurableSet.congr
    (ht : EventuallyMeasurableSet m l t) (hst : s =ᶠ[l] t) : EventuallyMeasurableSet m l s 
goal: EventuallyMeasurableSet m l s

full_name: EventuallyMeasurableSpace.measurableSingleton
code: instance measurableSingleton [MeasurableSingletonClass α] :
    @MeasurableSingletonClass α (EventuallyMeasurableSpace m l) 
goal: MeasurableSingletonClass α

full_name: EventuallyMeasurable
code: /-- We say a function is `EventuallyMeasurable` with respect to a given
σ-algebra `m` and σ-filter `l` if the preimage of any measurable set is equal to some
`m`-measurable set modulo `l`.
Warning: This is not always the same as being equal to some `m`-measurable function modulo `l`.
In general it is weaker. See `Measurable.eventuallyMeasurable_of_eventuallyEq`.
*TODO*: Add lemmas about when these are equivalent. -/
def EventuallyMeasurable (f : α → β) : Prop 
goal: Prop

full_name: Measurable.eventuallyMeasurable
code: theorem Measurable.eventuallyMeasurable (hf : Measurable f) : EventuallyMeasurable m l f 
goal: EventuallyMeasurable m l f

full_name: Measurable.comp_eventuallyMeasurable
code: theorem Measurable.comp_eventuallyMeasurable (hh : Measurable h) (hf : EventuallyMeasurable m l f) :
    EventuallyMeasurable m l (h ∘ f) 
goal: EventuallyMeasurable m l (h ∘ f)

full_name: EventuallyMeasurable.congr
code: /-- A function which is `EventuallyEq` to some `EventuallyMeasurable` function
is `EventuallyMeasurable`.-/
theorem EventuallyMeasurable.congr
    (hf : EventuallyMeasurable m l f) (hgf : g =ᶠ[l] f) : EventuallyMeasurable m l g 
goal: EventuallyMeasurable m l g

full_name: Measurable.eventuallyMeasurable_of_eventuallyEq
code: /-- A function which is `EventuallyEq` to some `Measurable` function is `EventuallyMeasurable`.-/
theorem Measurable.eventuallyMeasurable_of_eventuallyEq
    (hf : Measurable f) (hgf : g =ᶠ[l] f) : EventuallyMeasurable m l g 
goal: EventuallyMeasurable m l g