Question

Suppose (X, S,u) is a measure space and f1, f2, ... is a sequence of nonnegative S-measurable functions. Define f: X (0,00] by f(x) = =1 fx(x). Prove that s du = È I indu k=1

Answer 1

# Measure Theosy. ANN K-1 US - NEN Support (x, S, A) и 4 т т драи а fi, fa,.... is a sequence of non-negative S- measurable fintions . Define fix [0,00] by. f(x) = Ž for (x), then To prove that StoM= (s fada). Hoof! - Firstly we reall, the some important results. to prove . l It HalneN is non-negetice measurable, then. & fn = lim in is a meansable finehan. ® If f and g be two functions that are integrable Wirt. M. If os fix) = geag a. e. (4), then ffdus fg dh . 6 (Fatou's Lemma) let (ful be requence of measurable non-negative finctions. on the complete measure spall (X, S, M) and ta non negative function with time tn (x) = f(x) a. e (4). Then If de slim int sinde. No n no

| N N & P = = $ novo. V % * O Monotone convergence Theorem) het In) be a sequence of non-negative measurable functions with for cars fiti aie ca for everyn. Suppose tim fn(x) = feur 9. e. 4), then. Stal = line s fud M. Boof of (0) :- By Fatours lemma (2) Sfall slim inf Sindh By the definitions of Umint e limsup. Liman ffudle a lindup (fath Since the requence (hrine) is a non-decsensing sequence for almost every nex, for (x) a fex a e u fer alln. Hence by 6 Sindle e Sfdu. Consequently leon set (hn du S fdm. Putting these all together we Lave- Sfdes him in the In de alim dan south effth. Therfere State = sin intsande = line and Sande As a result in Schau exists and Sfde - line to lande This completes proof of 2 N B n nos

If f 8 g be non-negative measseble futions- For any nen-negative rumbss adb, S (af f6 g) du =affte +689d4. Now I am able to prove our main result. ZN No sco En is a meusnelle fubiony. n=1 Now by & , we have. El fondu) hormonu S. th an). De beste CE. cm dia) n=1 11 f. de a =ffidh. r ( de fon de) This tomy 11 다 This completes the