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In [[mathematics]] — specifically—specifically, in [[functional analysis]] — a—a '''weakly measurable function''' taking values in a [[Banach space]] is a [[function (mathematics)|function]] whose [[function composition|composition]] with any element of the [[continuous dual space|dual space]] is a [[measurable function]] in the usual (strong) sense. For [[separable space]]s, the notions of weak and strong measurability agree.
==Definition==
If (''X'', ΣΣ) is a [[measurable space]] and ''B'' is a Banach space over a [[field (mathematics)|field]] '''K''' (usually the [[real number]]s '''R''' or [[complex number]]s '''C'''), then ''f'' : ''X'' →→ ''B'' is said to be '''weakly measurable''' if, for every [[continuous linear functional]] ''g'' : ''B'' →→ '''K''', the function
:<math>g \circ f \colon X \to \mathbf{K} \colon x \mapsto g(f(x))</math>
is a measurable function with respect to ΣΣ and the usual [[Borel sigma algebra|Borel ''σσ''-algebra]] on '''K'''.
A measurable function on a [[probability space]] is usually referred to as a [[random variable]] (or [[random vector]] if it takes values in a vector space such as the Banach space ''B'').
Thus, as a special case of the above definition, if (ΩΩ, ΣΣ, '''P''') is a probability space, then a function ''Z'': : ΩΩ →→ ''B'' is called a (''B''-valued) '''weak random variable''' (or '''weak random vector''') if, for every continuous linear functional ''g'' : ''B'' →→ '''K''', the function
:<math>g \circ Z \colon \Omega \to \mathbf{K} \colon \omega \mapsto g(Z(\omega))</math>
is a '''K'''-valued random variable (i.e. measurable function) in the usual sense, with respect to ΣΣ and the usual Borel ''σσ''-algebra on '''K'''.
==Properties==
<blockquote>
A function ''f'' is said to be '''[[almost surely]] separably valued''' (or '''essentially separably valued''') if there exists a subset ''N'' ⊆⊆ ''X'' with ''μμ''(''N'') = 0 such that ''f''(''X'' \ ''N'') ⊆⊆ ''B'' is separable.
</blockquote>
<blockquote>
'''Theorem''' (Pettis)'''.''' A function ''f'' : ''X'' →→ ''B'' defined on a [[measure space]] (''X'', ΣΣ, ''μμ'') and taking values in a Banach space ''B'' is (strongly) measurable (with respect to ΣΣ and the Borel ''σσ''-algebra on ''B'') [[if and only if]] it is both weakly measurable and almost surely separably valued.
</blockquote>
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