Let \(\mathbf{X}\) be a \(d\)-dimensional random vector with distribution function \(F\). Let \(g: \mathbb{R}^{d} ightarrow \mathbb{R}\) be a continuous function such that \(|g(\mathbf{x})| \leq b\) for a finite real value \(b\) for all \(\mathbf{x} \in \mathbb{R}\). Let \(\varepsilon>0\) and define a partition of \([-b, b]\) given by \(a_{0}

\[\sum_{k=1}^{m}\left[\int_{A_{k}} g(\mathbf{x}) d F(\mathbf{x})-a_{k} P\left(\mathbf{X} \in A_{k}ight)ight] \geq-\varepsilon\]