Differentiation of measures (Q2744615)

In this article the authors proposed a new concept of differentiation of measures in the realm of \(C^1\) differentiable manifolds. The following is a description of the concept:NEWLINENEWLINENEWLINEFirst let \(\Omega\) be an open set of \(\mathbb{R}^n\) and \(\mu\) be a measure on \(\Omega\) taking a finite value on each compact subset. The authors say that \(\mu\) is differentiable at \(x\in\Omega\) if there exists a measure \(d_x\mu\) such that \(d_x\mu= \text{w-lim}_{t\to+0}\mu_t\), where \(\mu_t\) are the measures on \(\mathbb{R}^n\) defined by NEWLINE\[NEWLINE\mu_t(A)= \mu((x+ tA)\cap \Omega)/t^n.NEWLINE\]NEWLINE \(d_x\mu\) is said to be the differential of \(\mu\) at \(x\), and it is a measure on the tangent space \(T_x\Omega= \mathbb{R}^n\).NEWLINENEWLINENEWLINESeveral properties of the differential are discussed. In particular, the following statement is fundamental:NEWLINENEWLINENEWLINELet \(\varphi: U\to V\) be a \(C^1\) diffeomorphism between open subsets of \(\mathbb{R}^n\) and \(\mu\) be a measure on \(U\). If \(\mu\) is differentiable at a point \(x\in U\), then \(\varphi(\mu)\) is differentiable at \(y= \varphi(x)\) and we have \((d\varphi)(d_x\mu)= d_y(\varphi(\mu))\).NEWLINENEWLINENEWLINEIn virtue of the above theorem it is possible to introduce the notion of differentiation of measures on \(C^1\) differentiable manifolds \(X\) as measures on each tangent space \(T_xX\) in a natural way. This concept will be useful for a better understanding of the Radon-Nikodým derivative and the theory of conditional probability.