Remark to the ergodic decomposition of measures (Q752872)

Let (X,\({\mathfrak B},\mu)\) be a probability measure space and \({\mathfrak A}\) be a sub-\(\sigma\)-field of \({\mathfrak B}\). Consider a disintegration \(\{\mu^ x\}_{x\in X}\) of \(\mu\) with respect to \({\mathfrak A}\). (In general, such decomposition does not exist.) If the disintegration \(\{\mu^ x\}_{x\in X}\) satisfies a following condition (c), then it is called an ergodic decomposition. (c) For \(\mu\)-a.e. x, \(\mu^ x\) takes only the values 0 or 1 on \({\mathfrak A}.\) The following fact is known for the ergodic decomposition. Theorem. Let (X,\({\mathfrak B})\) be a standard space. \(\{\) \({\mathfrak A}_ n\}\) be a decreasing sequence of countably generated sub-\(\sigma\)-fields of \({\mathfrak B}\) and \({\mathfrak A}=\cap^{\infty}_{n=1}{\mathfrak A}_ n.\) Then for any probability measure \(\mu\) on \({\mathfrak B}\), the disintegration of \(\mu\) with respect to \({\mathfrak A}\) is ergodic. However even in a standard space, taking a suitable sub-\(\sigma\)-field \({\mathfrak A}\) there does exist a probability measure whose disintegration with respect to \({\mathfrak A}\) is non ergodic. The purpose of this note is to give such an example.