Conditional bounds and best \(L_{\infty}\)-approximations in probability spaces
The paper studies \(L_{\infty}\)-approximation in a probability space (\(\Omega\),\(\sigma\),P) by means of \(\alpha\)-measurable random variables for \(\alpha\) \(\subset \sigma\), a \(\sigma\)-lattice. Besides characterizing the set of best \(L_{\infty}\)-approximations, the case of their attainment has also been taken up. The helpful notion of `conditional bounds' developed in the paper leads one to stronger results and simpler proofs. The results imply that if the best \(L_{\infty}\)-approximation is unique it could be computed via the Pólya algorithm. The paper provides an alternative simpler-and-wider technique of proving the existing results in the area as also corrects a mistake in Lemma 4 of \textit{R. B. Darst} [Proc. Am. Math. Soc. 81, 433-436 (1981; Zbl 0461.41023)].
scientific article; zbMATH DE number 4140464
scientific article