Foundations — Probability space — sample space Ω, sigma-algebra F, measure P — Kolmogorov axioms

4.9.1 · D1 · Maths › Probability Theory & Statistics › Probability space — sample space Ω, sigma-algebra F, measure

Parent note Probability space padhne se pehle, tumhe har woh symbol pehchanna hoga jo woh use karta hai. Yeh page unhe zero se banata hai, aise order mein jahan har ek sirf usse pehle waale par depend karta hai.

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0. "Sets" ki alphabet (asli starting point)

Probability mein sab kuch sets se bana hai — cheezein ka collection. Toh hum wahan se shuru karte hain.

Figure s01 dekho: lavender blob ek set hai, har dot ek element hai. Arrow waala dot $x\in S$ satisfy karta hai; bahar waala dot $y\notin S$ satisfy karta hai. Yahi sab kuch symbol $\in$ ka matlab hota hai.

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1. Sets ko combine karna: complement, union, intersection

Yeh teen operations "events" ki poori grammar hain. Parent inhe hamesha use karta hai.

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2. Infinite lists aur symbols $\bigcup$, $\bigcap$, $\sum$

Parent do events par nahi rukta — woh infinitely many use karta hai. Teen "big operator" symbols yeh handle karte hain.

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3. Teen headline symbols: $\Omega$, $\mathcal{F}$, $P$

Ab parent note ke asli objects. Har ek Sections 0–2 se bana hai.

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4. Measurement symbols jo agle door milenge

Parent ke examples do aur notations par rely karte hain. Inhe abhi meet karo.

Recall Check: kya

$\mathcal{F}$ kabhi $2^{\Omega }$ se chhota hota hai? Haan ::: Uncountable $\Omega$ par (jaise $[0,1]$) non-measurable sets $\mathcal{F}$ (jaise Borel sigma-algebra) ko $2^{\Omega }$ ka strict subset banne par majboor karti hain.

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5. Foundations topic ko kaise feed karti hain

Map bottom-up padhta hai: raw set language $\Omega$, $\mathcal{F}$, aur $P$ banati hai; inhe bundle karne se triple milta hai; axioms upar rehte hain; aur wahan se baaki probability — Conditional probability & independence, Random variables, Expectation and Lebesgue integral — grow karti hai. Inclusion–exclusion principle aur Measure theory & Lebesgue integration usi machinery ko extend karte hain.

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Equipment checklist

Right-hand side cover karo aur khud test karo. Agar koi fuzzy lage, parent kholne se pehle uska section upar dobara padho.

• $x\in S$ ka matlab hai ::: $x$ set $S$ ka element hai (uske andar hai).
• $A\subseteq B$ ka matlab hai ::: $A$ ka har element $B$ mein bhi hai; $A$ poori tarah $B$ ke andar hai.
• $\varnothing$ hai ::: empty set — koi element nahi — impossible event ko model karta hai.
• $A^c$ hai ::: $A$ ka complement: $\Omega$ mein woh sab jo $A$ mein nahi hai ("not $A$").
• $A\cup B$ hai ::: union: $A$ ya $B$ ya dono mein outcomes ("kam se kam ek").
• $A\cap B$ hai ::: intersection: $A$ aur $B$ dono mein outcomes ("dono").
• "$A$ aur $B$ disjoint hain" ka matlab hai ::: $A\cap B=\varnothing$; koi outcome share nahi karte, isliye probabilities add ho sakti hain.
• ${\bigcup }_{i=1}^{\infty }{A}_i$ hai ::: ek infinite (countable) list ke kam se kam ek $A_i$ mein pade outcomes.
• ${\bigcap }_{i=1}^{\infty }{A}_i$ hai ::: ek saath saare $A_i$ mein pade outcomes.
• ${\sum }_{i=1}^{\infty }{p}_i$ hai ::: numbers ka running total $p_1+p_2+p_3+\cdots$.
• "countable" ka matlab hai ::: $1,2,3,\dots$ se ek ek karke label kiya ja sake, chahe list kabhi khatam na ho.
• $\Omega$ hai ::: sample space — saare possible outcomes $\omega$ ka set.
• $\mathcal{F}$ hai ::: sigma-algebra — $\Omega$ ke un subsets ka collection jinhe hum events kehte hain.
• $P:\mathcal{F}\to [0,1]$ ka matlab hai ::: $P$ ek event leta hai aur uski probability return karta hai, 0 se 1 inclusive tak ka ek number.
• $(\Omega ,\mathcal{F},P)$ hai ::: probability space — teen ingredients ek package ke roop mein bundle kiye gaye.
• $|A|$ hai ::: finite set $A$ mein elements ki sankhya.
• $2^{\Omega }$ hai ::: power set — $\Omega$ ke saare subsets; agar $|\Omega |=n$ toh $2^n$ members hain.