Theorems · Theorem · real analysis

eVariationOn.lowerSemicontinuous

∀ {α : Type u_1} [inst : LinearOrder α] {E : Type u_2} [inst_1 : PseudoEMetricSpace E] (s : Set α),
  LowerSemicontinuous fun f => eVariationOn f s

The map (eVariationOn · s) is lower semicontinuous for pointwise convergence on `s`. Pointwise convergence on s is encoded here as uniform convergence on the family consisting of the singletons of elements of s.

Defined in

Mathlib.Topology.EMetricSpace.BoundedVariation

Cited by

0 results in Mathlib

Foundations

Depth 151 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

LinearOrderPseudoEMetricSpace