Editing Null set (section)

==Haar null==

In a [[separable space|separable]] [[Banach space]] <math>(X, \|\cdot\|),</math> addition moves any subset <math>A \subseteq X</math> to the translates <math>A + x</math> for any <math>x \in X.</math> When there is a [[probability measure]] {{math|''μ''}} on the σ-algebra of [[Borel subset]]s of <math>X,</math> such that for all <math>x,</math> <math>\mu(A + x) = 0,</math> then <math>A</math> is a '''Haar null set'''.<ref>{{cite journal | first=Eva | last=Matouskova | date=1997 | url=https://www.ams.org/journals/proc/1997-125-06/S0002-9939-97-03776-3/S0002-9939-97-03776-3.pdf | title=Convexity and Haar Null Sets | journal=[[Proceedings of the American Mathematical Society]] | volume=125 | issue=6 | pages=1793–1799 | jstor=2162223| doi=10.1090/S0002-9939-97-03776-3 | doi-access=free }}</ref>

The term refers to the null invariance of the measures of translates, associating it with the complete invariance found with [[Haar measure]].

Some algebraic properties of [[topological group]]s have been related to the size of subsets and Haar null sets.<ref>{{cite journal | first=S. | last=Solecki | date=2005 | title=Sizes of subsets of groups and Haar null sets | journal=Geometric and Functional Analysis | volume=15 | pages=246–73 | mr=2140632 | doi=10.1007/s00039-005-0505-z| citeseerx=10.1.1.133.7074 | s2cid=11511821 }}</ref>
Haar null sets have been used in [[Polish group]]s to show that when {{mvar|A}} is not a [[meagre set]] then <math>A^{-1} A</math> contains an open neighborhood of the [[identity element]].<ref>{{cite journal | first=Pandelis | last=Dodos | date=2009 | title=The Steinhaus property and Haar-null sets | journal=[[Bulletin of the London Mathematical Society]] | volume=41 | issue=2 | pages=377–44 | mr=4296513| bibcode=2010arXiv1006.2675D | arxiv=1006.2675 | doi=10.1112/blms/bdp014 | s2cid=119174196 }}</ref> This property is named for [[Hugo Steinhaus]] since it is the conclusion of the [[Steinhaus theorem]].