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Mathlib.MeasureTheory.Measure.Tight

Tight sets of measures #

A set of measures is tight if for all 0 < ε, there exists a compact set K such that for all measures in the set, the complement of K has measure at most ε.

Main definitions #

Main statements #

def MeasureTheory.IsTightMeasureSet {𝓧 : Type u_1} [TopologicalSpace 𝓧] {m𝓧 : MeasurableSpace 𝓧} (S : Set (Measure 𝓧)) :

A set of measures S is tight if for all 0 < ε, there exists a compact set K such that for all μ ∈ S, μ Kᶜ ≤ ε. This is formulated in terms of filters, and proven equivalent to the definition above in IsTightMeasureSet_iff_exists_isCompact_measure_compl_le.

Equations
Instances For
    theorem MeasureTheory.IsTightMeasureSet_iff_exists_isCompact_measure_compl_le {𝓧 : Type u_1} [TopologicalSpace 𝓧] {m𝓧 : MeasurableSpace 𝓧} {S : Set (Measure 𝓧)} :
    IsTightMeasureSet S ∀ (ε : ENNReal), 0 < ε∃ (K : Set 𝓧), IsCompact K μS, μ K ε

    A set of measures S is tight if for all 0 < ε, there exists a compact set K such that for all μ ∈ S, μ Kᶜ ≤ ε.

    Finite measures that are inner regular with respect to closed compact sets are tight.

    Inner regular finite measures on T2 spaces are tight.

    In a complete second-countable pseudo-metric space, finite measures are tight.

    In a compact space, every set of measures is tight.

    theorem MeasureTheory.IsTightMeasureSet.subset {𝓧 : Type u_1} [TopologicalSpace 𝓧] {m𝓧 : MeasurableSpace 𝓧} {S T : Set (Measure 𝓧)} (hT : IsTightMeasureSet T) (hST : S T) :
    theorem MeasureTheory.IsTightMeasureSet.inter {𝓧 : Type u_1} [TopologicalSpace 𝓧] {m𝓧 : MeasurableSpace 𝓧} {S : Set (Measure 𝓧)} (hS : IsTightMeasureSet S) (T : Set (Measure 𝓧)) :
    theorem MeasureTheory.IsTightMeasureSet.map {𝓧 : Type u_1} {𝓨 : Type u_2} [TopologicalSpace 𝓧] {m𝓧 : MeasurableSpace 𝓧} {S : Set (Measure 𝓧)} [TopologicalSpace 𝓨] [MeasurableSpace 𝓨] [OpensMeasurableSpace 𝓨] [T2Space 𝓨] (hS : IsTightMeasureSet S) {f : 𝓧𝓨} (hf : Continuous f) :