Theorems · Theorem · general topology

tendsto_indicator_const_iff_forall_eventually

∀ {α : Type u_1} {A : Set α} {β : Type u_2} [inst : Zero β] [inst_1 : TopologicalSpace β] {ι : Type u_3} (L : Filter ι)
  {As : ι → Set α} [T1Space β] (b : β) [NeZero b],
  Filter.Tendsto (fun i => (As i).indicator fun x => b) L (nhds (A.indicator fun x => b)) ↔
    ∀ (x : α), ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A

The indicator functions of Asᵢ tend to the indicator function of A pointwise if and only if for every x, we eventually have the equivalence x ∈ Asᵢ ↔ x ∈ A.

Defined in

Mathlib.Topology.IndicatorConstPointwise

Cited by

2 results in Mathlib

Foundations

Depth 73 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

ZeroTopologicalSpaceT1SpaceNeZero