We study local interpolation properties and local supremum properties for a Boolean algebra. In particular, we present a new condition that is sufficient for the Nikodym property.

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 $\varphi (·,z_j\left(·\right))$ in X, where $z_j:\Omega \to {ℝ}^m$ is measurable for j ∈ ℕ and $\varphi :\Omega \times {ℝ}^m\to {ℝ}^d$ is a Carathéodory function. We obtain general sufficient conditions, under which, for some negligible set $A_{\varphi }$, the integral $I(\varphi ,{\nu }_x):={\int }_{{ℝ}^m}\varphi (x,\lambda )d{\nu }_x\left(\lambda \right)$ exists for $x\in \Omega \setminus A_{\varphi }$ and $\varphi ̅\left(x\right)=I(\varphi ,{\nu }_x)$ on $\Omega \setminus A_{\varphi }$, where $\nu ={{\nu }_x}_{x\in \Omega }$ is a measurable-dependent family of Radon probability measures on ${ℝ}^m$.