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

't is known that if you begin from Lebesgue point of view
you naturally end up with notation
where can be thought of as an infitesimal *set*;
Riemann on the other hand leads you to
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
is a differential form, that is it maps each
into a multilinear function -- a tangent,
a multilinear approximation of the integral curve.
Then even though the action of the linear operator
may involve limits in its definition -- which is perfectly fine --
our 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
"**for every** small enough a "
actually means that we **define a** linear **map** that turns **every**
into approximation of .

But why -- it feels like I fail to construct even for myself an analogous interpretation for all these or .

The naive way to describe the latter is to say simply
that it is a *random linear operator* -- the derivative of
for any fixed outcome .

Yet with the *set*... 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.

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