4.10.25 · D1Measure theory — Lebesgue measure (intro)
Hum number line ke point-sets ko ek "length" naam ka number dena chahte hain, jo ordinary intervals se agree kare aur weird, bikhari hui sets ke liye bhi kaam kare. Neeche har symbol ek kaam ke liye ek tool hai: ek set ko chhote intervals se cover karo, unki lengths add karo, aur total ko jitna ho sake utna chhota squeeze karo.
Is page mein kuch bhi assume nahi kiya gaya . Agar aapne number line dekhi hai aur fractions add kar sakte ho, toh yahan ki har line padh sakte ho. Hum parent note (Measure theory — Lebesgue measure (intro) ) mein aane wale har symbol se ek-ek karke milenge, aur koi bhi symbol use nahi karenge jab tak woh build na ho jaye.
R R sabhi numbers ki infinite seedhi line hai : whole numbers, fractions, aur beech ke endless wale jaise 2 π 0
Topic ko isko kyun chahiye. Jo bhi hum measure karte hain woh sab R andar rehta hai. Hamare liye "set" bas is line se chune gaye dots ka collection hai. Measure karne ka matlab hai: yeh dots ka collection kitni line occupy karta hai?
Intuition Ek "point" ek dot hai jiska koi width nahiEk single number, maano p = 0.5 zero width hai — aap sirf us dot ke aas-paas koi interval nahi fit kar sakte bina neighbours ko pakde. Yeh baat yaad rakhna: isi liye ek single point ka measure 0
⊆ ∅ set points ka collection hota hai. Hum A ⊆ R A R A ∅ empty set hai — ek bhi dot nahi is collection mein.
∪ ∩ E c
A ∪ B union ): woh saare dots jo A ya B A ∩ B intersection ): dots jo A aur B E c complement ): R E nahi hain — E
Topic ko isko kyun chahiye. Parent note mein Carathéodory test ek set A E A ∩ E E A ∩ E c grammar hain.
Definition Disjoint sets aur⨆ disjoint hain agar unmein koi common dot na ho (A ∩ B = ∅ ⨆ n E n pairwise disjoint hain" — yeh warn karta hai ki pieces overlap nahi karte, isliye unki lengths simply add ki ja sakti hain.
Definition Interval notation
[ a , b ] closed interval : a b a b including .( a , b ) open interval : strictly a b excluded .
Picture: number line par ek solid segment. Filled dot matlab "endpoint included", hollow dot matlab "excluded".
ℓ ℓ length hai: ℓ ( ( a , b ) ) = b − a ℓ ( [ a , b ] ) = b − a
Topic ko isko kyun chahiye. Intervals woh ekmaatra sets hain jinki length par hum pehle se agree karte hain. Hum jo bhi clever length compute karte hain woh set ko intervals se cover karke aur unki ℓ length nahi badalti (ek single endpoint zero-width dot hai) — isliye parent note covers mein freely open intervals use karta hai lekin answer μ ([ a , b ]) = b − a
Ek set countable hai agar aap uske dots ko ek list q 1 , q 2 , q 3 , … Q
Ek set uncountable hai agar aisi koi list kabhi bhi har dot ko pakad nahi sakti — unki counting numbers se strictly "zyada" hain. Poora interval [ 0 , 1 ]
Poori kahani ke liye Countable vs uncountable dekhein.
Intuition Yeh split kyun poore subject par raaj karta haiHaari measuring machine cover mein countably many intervals allow karti hai — list ke har item ke liye ek. Agar ek set countable hai, toh hum uske har dot ko apna ek chhota shrinking interval de sakte hain aur grand total ko jitna chahein utna chhota bana sakte hain. "Q [ 0 , 1 ] 1
∞ ∑ ∑ n = 1 ∞ x n x 1 + x 2 + x 3 + ⋯ converge karta hai), kabhi kabhi bina bound ke badhta rehta hai. ∞
Worked example Woh ek infinite sum jis par aapko trust karna hai∑ n = 1 ∞ 2 n 1 = 2 1 + 4 1 + 8 1 + ⋯ = 1. 1 paas jaate ho lekin kabhi overshoot nahi karte. Yeh geometric series parent note mein ε / 2 n
ε ε positive number ke liye stand karta hai jise aap jitna chahein utna chhota bana sakte ho . "For all ε > 0 "chahe aap kitni bhi tight tolerance demand karo, main use meet kar sakta hoon."
inf inf infimum ) woh sabse chhota value hai jis tak aap squeeze kar sakte ho — woh floor jis par numbers approach karte hain lekin shayad touch na karein.
Numbers ke dots ko line par imagine karo; inf
Example: intervals ( p − ε , p + ε ) 2 ε 0.2 , 0.02 , 0.002 , … 0 inf = 0
Intuition WHY infimum aur minimum nahi?Ek minimum ko kisi element dwara achieve kiya jana chahiye. Lekin ek set ka tightest cover often exist hi nahi karta — aap hamesha thoda aur shave kar sakte ho. Koi sabse chhoti positive length nahi hoti. Isliye hum inf "cover-totals kis value ki taraf crowd karte hain?" chahe kuch bhi exactly hit na kare. Yahi precise reason hai ki parent note ka outer measure inf
Topic ko dono kyun chahiye. Outer measure μ ∗ ( A ) define hi hota hai sabhi covers par inf ε → 0
Ab parent note ki headline definition ka har piece defined hai. Isko dheere padhein:
Ab aapke paas woh saare symbols hain jo parent note use karta hai. Isse aage kuch bhi assume nahi kiya gaya.
Union Intersection Complement
Infinite sum with geometric budget
Infimum squeezes the total
Lebesgue measure and measurable sets
Right side cover karo. Agar har ek ka jawab de sako, toh parent note ke liye taiyaar ho.
R Sabhi real numbers ki infinite number line.
Ek "point" kya hai aur uski width kitni hai? Line par ek single number / dot, zero width ke saath.
A ⊆ B A B A B
E c Complement — R E
⨆ n E n ⋃ Pieces pairwise disjoint hain (koi overlap nahi), isliye lengths simply add ki ja sakti hain.
Interval ( 3 , 7 ) ℓ 7 − 3 = 4
Kya endpoints include ya exclude karne se interval ki length badalti hai? Nahi — ek single endpoint ek zero-width point hai.
Ek set ke countable hone ka kya matlab hai? Uske elements ko q 1 , q 2 , q 3 , …
∑ n = 1 ∞ 2 n 1 1
"For all ε > 0 Aap koi bhi positive tolerance meet kar sakte ho, chahe kitni bhi tiny ho.
Outer measure mein min inf Tightest cover kabhi achieve nahi ho sakta; inf
A ⊆ ⋃ n I n I n Woh open intervals hain jo A
