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 #
MeasureTheory.IsTightMeasureSet
: A set of measuresS
is tight if for all0 < ε
, there exists a compact setK
such that for allμ ∈ S
,μ Kᶜ ≤ ε
. The definition uses an equivalent formulation with filters:⨆ μ ∈ S, μ
tends to0
along the filter of cocompact sets.isTightMeasureSet_iff_exists_isCompact_measure_compl_le
establishes equivalence between the two definitions.
Main statements #
isTightMeasureSet_singleton_of_innerRegularWRT
: every finite, inner-regular measure is tight.
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
- MeasureTheory.IsTightMeasureSet S = Filter.Tendsto (⨆ μ ∈ S, ⇑μ) (Filter.cocompact 𝓧).smallSets (nhds 0)
Instances For
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.