ba space

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the ba space 
 
 
 
 b
 a
 (
 Σ
 )
 
 
 {\displaystyle ba(\Sigma )}
 
 of an <a href="/facts/Field_of_sets/4E3vQl4f">algebra of sets</a> 
 
 
 
 Σ
 
 
 {\displaystyle \Sigma }
 
 is the <a href="/facts/Banach_space/sDEIk7EC">Banach space</a> consisting of all <a href="/facts/Bounded_measure/whVjbtuk">bounded</a> and finitely additive <a href="/facts/Signed_measure/vukVf84A">signed measures</a> on 
 
 
 
 Σ
 
 
 {\displaystyle \Sigma }
 
. The norm is defined as the <a href="/facts/Measure_variation/tyY5fTz9">variation</a>, that is 
 
 
 
 ‖
 ν
 ‖
 =
 
 |
 
 ν
 
 |
 
 (
 X
 )
 .
 
 
 {\displaystyle \|\nu \|=|\nu |(X).}

If Σ is a <a href="/facts/Sigma-algebra/8LJINLNc">sigma-algebra</a>, then the space 
 
 
 
 c
 a
 (
 Σ
 )
 
 
 {\displaystyle ca(\Sigma )}
 
 is defined as the subset of 
 
 
 
 b
 a
 (
 Σ
 )
 
 
 {\displaystyle ba(\Sigma )}
 
 consisting of <a href="/facts/Sigma-additive/gpgoKIeU">countably additive measures</a>. The notation ba is a <a href="/facts/Mnemonic/6HfDn0ks">mnemonic</a> for bounded additive and ca is short for countably additive.
If X is a <a href="/facts/Topological_space/v2ut7CbK">topological space</a>, and Σ is the sigma-algebra of <a href="/facts/Borel_set/jvSsfpkP">Borel sets</a> in X, then 
 
 
 
 r
 c
 a
 (
 X
 )
 
 
 {\displaystyle rca(X)}
 
 is the subspace of 
 
 
 
 c
 a
 (
 Σ
 )
 
 
 {\displaystyle ca(\Sigma )}
 
 consisting of all <a href="/facts/Regular_measure/zqVDoMv4">regular</a> <a href="/facts/Borel_measure/8cYR8Ysw">Borel measures</a> on X.

ba space open-in-new

ba space