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This post will be essentially about functions of bounded variation of one variable. The main source is the book “Functions of Bounded variation and Free Discontinuity Problems” by Ambrosio, Fusco and Pallara. Before we give the definition of a bounded variation function let us recall what exactly does is mean for a function to belong in . Recall that any function can be seen as a distribution i.e. as a bounded linear functional on , with
In that case we say that the distribution is representable by the function . Given any distribution we can define its distributional derivative to be the distribution defined as
In the special case where the distribution can be represented by a function in the way we show above the distributional derivative will be
There is a remarkably nice proof of the Lebesgue decomposition theorem (described below) by von Neumann. This leads immediately to the Radon-Nikodym theorem.
If and are two finite measures on then there exists a non-negative (w.r.t. both measures) measurable function and a -null set such that
for each .
Let and consider the operator
The fundamental theorem of algebra states that is algebraically closed, that is;
For any non-constant polynomial in , there exists a such that .
Let be a Brownian motion on and suppose for a contradiction that a non-constant polynomial does not have any zero’s. Let , then is analytic and tends to 0 at infinity. Pick such that and note that and contain an open set, which can be done due to the fact that is continuous and non-constant.
Now is a continuous local martingale (by using Ito’s formula) and moreover it is bounded. Hence by the Martingale convergence we have that a.s. and in .
This last statement is contradicted by the fact that Brownian motion is recurrent on the complex plane, in particular, it visits and infinitely many times which gives that
directly contradicting the Martingale convergence.
I found this little gem in Rogers and Williams.