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Let 〈X,Y 〉 be a duality pair of M -spaces X,Y of measurable functions from Ω ⊂ R into R. The paper deals with Y -weak cluster points φ of the sequence φ(·, zj(·)) in X, where zj : Ω → R is measurable for j ∈ N and φ : Ω×R → R is a Carathéodory function. We obtain general sufficient conditions, under which, for some negligible set Aφ, the integral I(φ, νx) := Rm φ(x, λ) dνx(λ) exists for x ∈ Ω \ Aφ and φ(x) = I(φ, νx) on Ω \ Aφ, where ν = {νx}x∈Ω is a measurable-dependent family of Radon probability measures on R.

The Young Measure Representation for Weak Cluster Points of Sequences in M-spaces of Measurable Functions