this book rewired my brain.
Øksendal's book on SDEs. heavy. theoretical. it'll take you weeks, maybe months. but it's the cleanest path i've found from measure theory into real stochastic analysis.
Brownian motion gets built properly. then the itô integral. then itô's formula. then diffusions. each one earned from what came before.
the funny part is the subtitle says "with applications." it isn't really. you spend chapters on martingales before you see a single financial quantity. semigroups and generators sit at the core. optimal stopping shows up before you ever price an option. black-scholes is on the last few pages and almost incidental.
worth every hour. you finish it and stochastic processes stop feeling like magic. they feel like geometry.
draw any loop on a circle. that loop is secretly an integer.
take a circle, pick a point, and draw a loop that starts and ends there. you can describe the whole loop by how many times it wraps around on its way back. once clockwise is +1. twice is +2. counterclockwise gives negatives. no wrap is 0.
here's the move. in topology, two loops are "the same" if you can smoothly deform one into the other. and once you allow that, everything about the loop washes out except the winding number. how wiggly it is, where it bunches up, none of it survives. only the wrap count.
so the set of essentially different loops on a circle is just the integers. an infinite topological universe collapses to ℤ.
this is the fundamental group of the circle. π_1(S¹) ≅ ℤ. one of the cleanest theorems in topology, because it takes a continuous question (which loops can deform into which) and gives a purely discrete answer (count the wraps).
a sequence of functions can converge to f in Lp without converging to f at a single point.
classic example: the typewriter sequence. bumps that slide across [0, 1] with shrinking width but constant height. the L¹ norm collapses to 0. but at any fixed x, f_k(x) keeps flipping between 0 and 1 forever.
Lp convergence isn't about points. it's about the norm collapsing. the topology forgets local behavior and only remembers integrals.
what's actually satisfying is how much theory shows up the moment you ask why.
the unit ball of Lp isn't compact in the norm topology. that's a functional analysis fact, and it's why you can't extract a pointwise convergent subsequence in general. but the riesz subsequence theorem rescues something: given any Lp convergent sequence, there's a subsequence that converges almost everywhere. a faint trace of compactness restored by measure theory.
and then banach-alaoglu shows up to finish the job. norm compactness fails, but the closed unit ball is compact in the weak-* topology. you lose pointwise control. you keep geometric control. just in a weaker form.
real analysis, measure theory, functional analysis, topology, and a piece of geometry, all hiding inside a single statement: f_k → f in Lp.
one definition. five subjects answering it. that's why functional analysis feels like the spine of modern math.
spent today proving radon-nikodym.
if one measure is absolutely continuous with respect to another, there's a function h that links them. integrating against h recovers the first from the second. it's the formal way to say "this measure is just that one, reweighted".
without this, conditional expectation isn't a real mathematical object. probability collapses.
the beautiful part is the move you don't expect. you don't construct h directly. you combine both measures into a sum, work in the hilbert space L^2 over that sum, and define a linear functional that integrates against the first measure. riesz representation hands you a function g. a few indicator tricks show 0 <= g < 1 almost everywhere. and the h you wanted falls out as g/(1-g).
you never build it. the hilbert space geometry summons it.
probability rests on a riesz argument applied to the right space.
P(A) = 1 doesn't mean A is guaranteed.
if you pick a number uniformly between 0 and 1, the probability it's irrational is 1, and the probability it's rational is 0. but it's still possible to pick a rational. it just happens with probability 0.
probability 1 doesn't mean certain. it means "almost surely". the exceptions exist, they just live on a set of measure zero, which probability theory treats as negligible.
even certainty comes with a margin of measure 0.
