[數學普及] Measure Theory (測度論) 簡介 ver. 2

咁搞法分分鐘連數學本科都唔明

measure theory 同太多field 有connections, 真係好難有 “the” way to teach. 可能同小弟background 有關, 我覺得用 l^p, Banach, Hilbert spaces in finite dimensional case 去 illustrate 部份measure theory concepts 其實幾幫到手, 有時比Riemann-Lebesgue 嗰view清晰d

grad course都唔會好似你咁教啦

小弟最初接觸measure theory 係讀grad course 嗰時, Rudin直入, 由functional analyst 教; 頭幾個chapters 辛苦到爆, 連Lebesgue integration 都係教完Reisz 後當corollary. 雖然好抽象但對後期同其他field接軌時個big picture 宏觀好多

呢個approach只適合advanced graduates
嚴重離地

可能大家focus位唔同

同埋我用extension 多過 riesz...

了解, 我詳睇返師兄前面d notes, 感覺如果後面你要講返點解 Lebesgue integral 真係可以 extend 到 Riemann integral (三大問題, Borel sets), 最後原點都一定要cover返 Borel sets 同部份 basic topology on R or R^n.

real line 易搞好多, 基本上知open set = disjoint union of countably many open intervals 就ok
somehow我唔打算講 R^n, n > 1, 我寧願當product measure算...

R^n 唔係都係open ball gen出黎咩？

R^n個Lebesgue measure可以係用open ball砌 (as outer measure), 亦都可以用n 個 one-dimensional ge Lebesgue measure 去砌 (as a product measure)
呢D執過前面堆文之後會再講

之後有一條大定理叫 caratheodory extension theorem, 條theorem話比我地知兩個measure都係一樣

wa, 強帖留名...

https://lihkgmeasure.wordpress.com/
砍掉重練版

留名

無野既...

而家再望下

開頭其實基本上係一樣, 係之後變左

我讀果間垃圾U得applied math,無呢d pure math讀

假設f(x)有一interval [a,b]
我要點樣唔用in symbol表達呢個interval既area
而用sigma黎表達佢?

因為in呢個符號我比較難直接聯想返積分個基本concept
用sigma黎表達the sum of Infinitesimal
可能會比較易理解integration個概念

就算得applied math 都應該有measure theory架喎，基礎野黎

得probability modelling果d
set果d算唔算？

Probability 係 measure theoretic?