EN
Let ⟨X,Y⟩ be a duality pair of M-spaces X,Y of measurable functions from Ω ⊂ ℝ ⁿ into $ℝ^d$. The paper deals with Y-weak cluster points ϕ̅ of the sequence $ϕ(·,z_{j}(·))$ in X, where $z_j:Ω → ℝ^m$ is measurable for j ∈ ℕ and $ϕ:Ω×ℝ^m → ℝ^d$ is a Carathéodory function. We obtain general sufficient conditions, under which, for some negligible set $A_ϕ$, the integral $I(ϕ,ν_x):= ∫_{ℝ^m} ϕ(x,λ) dν_x(λ)$ exists for $x ∈ Ω∖ A_ϕ$ and $ϕ̅(x) = I(ϕ,ν_x)$ on $Ω∖ A_ϕ$, where $ν={ν_x}_{x ∈ Ω}$ is a measurable-dependent family of Radon probability measures on $ℝ^m$.