There are beautiful math, there are ugly math, and there are ugly math, whose elengance can only be appreciated after suffering.

Zahl presenting one of the most important results in analysis over the past 20 years — The Kakeya Conjecture

Today I am reading Peres and Pörter's Brownian motion. In the part of deriving the Frostman's lemma, the original context states that $A\subset\mathbb{R}^d$ is a closed set while in other contexts $A$ is assumed to be compact. I reached to Deepseek and this prized goofy states this...

The Itô's isometry yields

To show this equality holds, we first square both sides and then take expectations on both sides, then we have

We wish to show that $\mathcal{H}$ is a complete subspace of $(\Omega,\mathcal{F}_\infty,\mathbb{P})$ and then by the Hilbert space structure it is closed. To show that we need the equaltiy

I am recently reading LeGall's Brownian motion. There is a step in proving Martingale Representation Theorem (Theorem 5.18) I don't understand (see Figure 1). The idea in proving assertion (i) is to use $\mathcal{H}$ to denote all such $Z$ in $L^2$, then prove it is a closed subspace (see Figure 2).