On adjointness of a group action to its inverse (Q913970)

Let \({\mathfrak U}(X,\mu)\) be the group of automorphisms of the Lebesgue space (X,\(\mu\)). The author considers a measurable action T of the group \({\mathbb{Z}}^ k\) as a set \((T_ 1,T_ 2,...,T_ k),\) where \(T_ i\in {\mathfrak U}(X,\mu)\) and \(T_ iT_ j=T_ jT_ i\) for \(i,j\in (1,...,k).\) He considers \({\mathfrak U}(X,\mu,{\mathbb{Z}}^ k)\) as a subset of \(\prod^{k}_{i=1}{\mathfrak U}(X,\mu)\) and defines on the \({\mathfrak U}(X,\mu,{\mathbb{Z}}^ k)\) the weak topology as induced topology from \(\prod^{k}_{i=1}{\mathfrak U}(X,\mu).\) Theorem: The set of measurable actions of the group \({\mathbb{Z}}^ k\), nonadjoint to its inverse, contains a dense subset of type \(G_{\delta}.\) For the case \(k=1\) this theorem was proved by A. del Junco.