Bochner measurable function

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a> – specifically, in <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a> – a Bochner-measurable function taking values in a <a href="/facts/Banach_space/sDEIk7EC">Banach space</a> is a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> that equals almost everywhere the limit of a sequence of measurable countably-valued functions, i.e.,

f
        (
        t
        )
        =
        
          lim
          
            n
            →
            ∞
          
        
        
          f
          
            n
          
        
        (
        t
        )
        
           for almost every 
        
        t
        ,
        
      
    
    {\displaystyle f(t)=\lim _{n\rightarrow \infty }f_{n}(t){\text{ for almost every }}t,\,}

where the functions 
 
 
 
 
 f
 
 n
 
 
 
 
 {\displaystyle f_{n}}
 
 each have a countable range and for which the pre-image 
 
 
 
 
 f
 
 n
 
 
 −
 1
 
 
 (
 {
 x
 }
 )
 
 
 {\displaystyle f_{n}^{-1}(\{x\})}
 
 is measurable for each element x. The concept is named after <a href="/facts/Salomon_Bochner/n35gmr7h">Salomon Bochner</a>.
Bochner-measurable functions are sometimes called <a href="/facts/Strongly_measurable/RknLPSK9">strongly measurable</a>, 
 
 
 
 μ
 
 
 {\displaystyle \mu }
 
-measurable or just measurable (or uniformly measurable in case that the Banach space is the space of continuous <a href="/facts/Linear_operator/Q1PBKNVV">linear operators</a> between Banach spaces).

Bochner measurable function open-in-new

Bochner measurable function