An aspect of differentiable measures on \({\mathbb{R}}^{\infty}\) (Q1106536)

Let \(R^{\infty}\) be a countable direct product of \(R^ 1\), \({\mathcal B}(R^{\infty})\) be the usual Borel \(\sigma\)-algebra on \(R^{\infty}\) and \(\mu\) be a probability measure on \({\mathcal B}(R^{\infty})\). \(\mu\) is said to be differentiable in the a-direction, or a is said to be a differentiable shift, if \[ \lim_{h\to 0}h^{-1}\{\mu (E-ta)-\mu (E)\}\equiv \partial_ a\mu (E) \] exists for all \(E\in {\mathcal B}(R^{\infty})\). In this case \(\partial_ a\mu\) is absolutely continuous with respect to \(\mu\). So the set \(D_{\mu}\) of all differentiable shifts for \(\mu\) equipped with the norm \[ \| a\|_{\mu}=\int | d\partial_ a\mu /d\mu (x)| d\mu (x) \] is a Banach space of cotype 2. We investigate \(R_ 0^{\infty}\)- differentiable measures \(\mu\), especially direct-product measures \(\prod_{n}\mu_ n\) of \(\mu_ n\), being 1-dimensional probability measures on \({\mathcal B}(R^ 1)\), where \[ R_ 0^{\infty}=\{a=(a_ n)\in R^{\infty}| a_ n=0\text{ except for a finite number of n's}\}. \] The following main results are obtained. (1) Let \(\mu =\prod_{n}\mu_ n\) be \(R_ 0^{\infty}\)-differentiable on \(R^ 1\), \(\mu_ n\) \((n=1,...)\) have the density \(f_ n(x)\) with respect to the Lebesgue measure dx on \(R^ 1\) and \(f_ n'(x)\) belong to \(L^ 1_{dx}(R^ 1)\). Put \(\phi_ n(x)=f_ n'/f_ n(x)\). Then \[ D_{\mu}=\{a=(a_ n)\in R^{\infty}| \sum_{n}M_ n(a_ n)<\infty \}, \] where \(M_ n\) \((n=1,...)\) is an Orlicz function on \(R^ 1\) defined by \[ M_ n(\alpha)=\int^{\infty}_{- \infty}dx\int^{1}_{0}\{1-\exp (-2^{-1}\alpha^ 2u^ 2\phi^ 2_ n(x))\}u^{-2}du. \] Thus \(D_{\mu}\) is a modular sequence space. (2) In particular, if \(\mu\) is a stationary product, that is \(\mu_ n\) is the same measure, say \(\mu_ 0\) for all n, then \(D_{\mu}\) is an Orlicz sequence space defined by the function \(M_ 0(\alpha)\). So we can give necessary and sufficient conditions for the problem if there exists some \(\mu\) such that \(D_{\mu}=\ell_ M\), or we can give a characterization of \(\mu\) such that \(D_{\mu}=\ell_ M\) for given Banach spaces \(\ell_ M.\) (3) Lastly we obtained some results related to translational-quasi- invariance.