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), reach out!

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

Homework 1 (First draft)

Homework 2

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.