# $\mu(\mathrm{d}x)$ and $\mathrm{d}\mu(x)$

't is known that if you begin from Lebesgue point of view you naturally end up with notation $\mu(\mathrm{d}x)$ where $\mathrm{d}x$ can be thought of as an infitesimal set; Riemann on the other hand leads you to $\mathrm{d}\mu(x)$ which stands for "change of measure". This is all of course a birdish language which doesn't define anything for real and does not help computation.

Now, I really hate all these limit-centric notations. In smooth and bounded non-random finite-dimensional Euclidean world (in other words in any naive calculus) we most elegantly get rid of it, saying that $\mathrm{d}x$ is a differential form, that is it maps each $x$ into a multilinear function -- a tangent, a multilinear approximation of the integral curve. Then even though the action of the linear operator $\int_A$ may involve limits in its definition -- which is perfectly fine -- our $\mathrm{d}x$ is a pretty understandable object that has specific type. The integral curve is naturally characterized as the one with given initial value and tangents at each point. This in a rigorous way captures the original intuition of operating with "infinitesimal increments". The bird-language phrasing "$\mathrm{d}f \approx \frac{\mathrm{d}f}{\mathrm{d}x}\mathrm{d}x$for every small enough a $\mathrm{d}x$" actually means that we define a linear map that turns every $\Delta x$ into approximation of $\Delta f$.

But why -- it feels like I fail to construct even for myself an analogous interpretation for all these $\mu\mathrm{d}x$ or $\mathrm{d}W_t$.

The naive way to describe the latter is to say simply that it is a random linear operator -- the derivative of $t\mapsto W_t(\omega)$ for any fixed outcome $\omega$.

Yet with $\mathrm{d}x$ the set... $2^X$ isn't very linear a space and it isn't about linearity neither. Also this notation isn't going in line with the main idea -- that we're actually splitting the image space instead of the domain.