Differentiation of integrals in \(\mathbb R^{\omega}\) (Q5933724)

Let \(\lambda^{\omega}\) denote a version of ``Lebesgue measure'' in the separable Fréchet space \({\mathbb R}^{\omega}\) [see \textit{R. Baker}, Proc. Am. Math. Soc. 113, No. 4, 1023--1029 (1991; Zbl 0741.28009)]. The author proves that if \(f\in L^1({\mathbb R}^{\omega}, \lambda^{\omega})\), then for almost every point with respect to \(\lambda^{\omega}\) we have \[ {\lim}(\int_{x+C}f\;d\lambda^{\omega})/(\lambda^{\omega}(C))=0, \] where \(C\) is a rectangle centered at zero with \(0<\lambda^{\omega}(C)<\infty\), and taking the limit means that \(\operatorname{diam}(C)\to 0\). This shows that the Lebesgue differentiation theorem does not hold in this case.