Lecture Notes

Hi everyone, I'm Rasmus and I'm an exchange student all the way from Denmark. Back home I study Machine Learning and have done research in the field of NLP, specifically studying multilingual models. I'm at Berkeley for a semester to study mathematics for a semester in order to get more familiar with the rigorous nature of maths (ML research is basically result driven with little theory to back it up - godspeed!). If you want to hear about whether AI will kill us all one day (It might, but then again so will global warming. edit: or Russia. Слава Україні!)

I'm gonna publish my notes on the lectures this weekend after I finish this weeks homework.

Homework 1

Homework 2

Homework 3

Homework 4

Resume of the Lebesque Integral We begin by covering the Riemann integral in short. Riemann integration, like all others, seek to measure the area under a graph. It does this using approximation by partitioning the area into rectangles. In this function the Riemann sum corresponds to $$ \sum_{i=1}^nf(t_i)(x_i-x_{i-1}) $$ Letting $\Delta_{x_i}=x_i-x_{i-1}$ be the same value for all $i$, and letting $x_{i-1}\leq t_i\leq x_i\forall i$. If $a<x_i<b\forall i$, then we can define the Riemann integral as $$ \int_a^b f(x) dx = \lim_{n\to\infty}\sum_{i=1}^nf(t_i)(x_i-x_{i-1} $$ If the limit of this sum exists and is unique, then the function $f$ is Riemann integrable.

The Lebesque integral is based on the measurability of a functions undergraph. This undergraph is defined by $$ Uf=\{(x,y)\in\mathbb{R}\times[0,\infty):p\leq y<f(x)\} $$ We say that the function $f$ is Lebesque measurable if the undergraph $Uf$ is measurable. If $f$ is Lebesque measurable, then we let $$ \int f=m(Uf) $$ Note that $dx$ is missing. Because the Lebesque integral does not handle a limit of a sum of rectangles with width of $\Delta x_i$, we omit the $dx$.

Finally, a function is Lebesque integrable if the measure $m(Uf)$ is finite. Since the Lebesgue measure of $Uf$ can be infinity, we do by definition allow the Lebesque integral of $f$ to be infinite.

Homework 5

Resume of Lebesque measure theory

*This is gonna be a rough summarisation of all we've covered in Lebesque measure theory so far. It will probably not contain everything of importance and might have some gaps that I didn't think to cover. If you find gaps like these, please do write to me on Discord so I can review these. Thanks!*

Our venture into Lebesque measure theory begins with the definition of the outer measure - a measure of a subset $A$ found by $$ m^*(A)=\inf\left\{ \sum_k|B_k|:\{B_k\} \text{ is a covering of } A \text{ by open boxes} \right\} $$ Useful from this definition and this section is the definition that sets with outer measure zero is called a zero set. Boxes are created from intervals $(a_i,b_i)$ since the measure of an interval is its end point minus its starting point.

A set $A\subset R$ is then measurable if it the division $A|A^c$ is so *clean* that for all subsets $X\subset R$, $$ m^*(A)=m^*(X\cap A) + m^*(A\cap E^c) $$ Although I personally like the Tao condition better: $$ m^*(A)=m^*(X\cap A) + m^*(A\setminus E) $$ Throughout the course we have proved a lot of properties of measurable sets, including sub-additivities of outer measures, monotonicity, etc.