Thm: Let ${ X }$ be a set with ${ \sigma -}$algebra ${ \mathfrak{M} }.$ Let ${ f : X \to [0, \infty] }$ be a measurable map. Then there exist simple measurable maps ${ 0 \leq s _1 \leq s _2 \leq \ldots (\leq f) }$ with pointwise limit ${ \lim _{n \to \infty} s _n (x) = f(x) }$ for all ${ x \in X }.$ Pf: We can try forming a sequence ${ \varphi... Read more 12 Dec 2023 - 1 minute read
Ref: Prof. Vershynin’s handwritten notes Lec-1: Consider the problem of numerically computing the integral of an ${ f : [0,1] ^d \to \mathbb{R} }.$ Breaking ${ [0,1] ^d }$ into (axis aligned) cubes of width ${ \epsilon },$ there are about ${ N \approx (\frac{1}{\epsilon}) ^{d} }$ many such smaller cubes. Now the integral ${ \int _{[0,1] ^d} f... Read more 07 Oct 2023 - 23 minute read
Link to lectures Instructor: Prof. Philippe Rigollet ROUGH NOTES (!) Lec-1 [Slides] Idea: Use data to get insight and perhaps make decisions. Computational view: Data is a (large) sequence of numbers that needs to be processed by a fast algorithm (approximate nearest neighbours, low dimensional embeddings, etc) Statistical view: Data comes... Read more 31 Oct 2022 - 11 minute read
Let ${ f \in \mathcal{C}[a,b] }.$ Weierstrass approximation theorem says there is a sequence of polynomials uniformly converging to ${ f }$ on ${ [a,b] }.$ That is, for every ${ \epsilon \gt 0 }$ there is a polynomial ${ P }$ with ${ \max _{x \in [a,b]} \vert f (x) - P(x) \vert }$ ${ \lt \epsilon }.$ The proof below is originally due to Lebesg... Read more 22 Jan 2022 - 7 minute read
(Ref: Stromberg’s “Classical Real Analysis”) Let ${ \alpha \in \mathbb{R} }.$ The generalised binomial theorem says ${ (1+x) ^{\alpha} }$ ${ = \sum _{0} ^{\infty} \binom{\alpha}{j} x ^j }$ for ${ \vert x \vert \lt 1 }.$ Also, radius of convergence of ${ \sum _{1} ^{\infty} \binom{\alpha}{j} x ^j }$ is ${ \infty }$ when ${ \alpha \in \mathbb{... Read more 15 Jan 2022 - 4 minute read