Measure theory — Lebesgue measure (intro) — Hinglish notes

4.10.25 · Maths › Advanced Topics (Elite Level)

WHY hame yeh chahiye hi?

WHAT hum chahte hain — "length" $μ$ ki wishlist:

$μ ([a, b]) = b - a$ (intervals se agree kare)
$μ (\emptyset) = 0$ , $μ \geq 0$
Countable additivity: disjoint $A_{1}, A_{2}, \dots \Rightarrow μ (⋃ A_{n}) = \sum μ (A_{n})$
Translation invariance: $μ (A + t) = μ (A)$

Yeh nikla ki $R$ ke har subset ke liye yeh charon ek saath nahi ho saktin (Vitali set ise tod deta hai). Fix yeh hai: sirf ek badi class ke "nice" sets ko measure karo — jinhe measurable sets kehte hain.

HOW hum ise build karte hain: pehle outer measure

Basic properties scratch se derive karna

(P1) Monotonicity: agar $A \subseteq B$ toh $μ^{*} (A) \leq μ^{*} (B)$ . Kyun? $B$ ka har cover $A$ ka bhi cover hai. Toh $A$ ke cover-sums ka set $B$ ke cover-sums ka set contain karta hai; bade set par inf $\leq$ hota hai. Done.

(P2) Countable sub-additivity: $μ^{*} (⋃_{n} A_{n}) \leq \sum_{n} μ^{*} (A_{n})$ . Kyun & Kaise ( $ε / 2^{n}$ trick): Fix karo $ε > 0$ . Har $A_{n}$ ke liye ek cover ${I_{n, k}}_{k}$ chuno jisme $\sum_{k} ℓ (I_{n, k}) \leq μ^{*} (A_{n}) + \frac{ε}{2 ^{n}} .$ Yeh step kyun? inf ki definition se, koi cover $μ^{*} (A_{n})$ ko kisi bhi margin se beat karta hai; hum margin $ε / 2^{n}$ lete hain taaki errors ek geometric series banayein. Combined family ${I_{n, k}}_{n, k}$ ek countable cover hai $⋃_{n} A_{n}$ ka, toh $μ^{*} (⋃_{n} A_{n}) \leq \sum_{n, k} ℓ (I_{n, k}) \leq \sum_{n} (μ^{*} (A_{n}) + \frac{ε}{2 ^{n}}) = \sum_{n} μ^{*} (A_{n}) + ε .$ $ε \to 0$ karo. $■$

(P3) $μ^{*} ([a, b]) = b - a$ .

" $\leq$ ": akela cover $(a - ε, b + ε)$ ki length $b - a + 2 ε$ hai, toh $μ^{*} \leq b - a$ .
" $\geq$ " subtle direction hai (compactness chahiye): compact $[a, b]$ ke kisi bhi open cover ka finite subcover hota hai, jisski lengths ka total $\geq b - a$ hona chahiye (warna koi gap uncovered reh jaayega). Toh $μ^{*} \geq b - a$ . Milaakar: $= b - a$ . ✔ wishlist item 1 se match karta hai.

Figure — Measure theory — Lebesgue measure (intro)

HOW hum "nice" sets chunte hain: Carathéodory criterion

Worked examples

Common mistakes (steel-manned)

Active recall

Recall Pehle khud try karo

"Length" ke liye chahiye 4 properties ki wishlist batao.
$μ^{*} (A)$ scratch se define karo.
$μ^{*} (Q) = 0$ kyun? ( $ε / 2^{n}$ argument do.)
Carathéodory ki measurability criterion batao aur explain karo kyun split exactly add up karni chahiye.
Kaun si property sub-additive (saare sets) se additive (measurable sets) upgrade hoti hai?

What is Lebesgue outer measure

μ^{*} (A)

Open intervals se

A

ke saare countable covers par

\sum_{n} ℓ (I_{n})

ka infimum — sabse tight total cover length.

Why is

μ^{*}

only sub-additive, not additive, on all sets?

Pathological sets (jaise Vitali set) measure "leak" kar sakte hain; full additivity sirf measurable sets par

σ

-algebra

M

mein hoti hai.

State Carathéodory's measurability criterion.

E

measurable hai iff har test set

A

ke liye

μ^{*} (A) = μ^{*} (A \cap E) + μ^{*} (A \cap E^{c})

What is

μ^{*} (Q)

and why?

0

;

n

-ve rational ko length

ε / 2^{n}

ke interval se cover karo, total

\leq ε \to 0

. Countable sets null hote hain.

What is

μ ([a, b])

and which direction needs compactness?

b - a

; lower bound "

\geq

" compact

[a, b]

ke finite subcover use karta hai.

Which key property fails for Riemann but holds for Lebesgue?

Countable additivity of size/integral — wild functions jaise

1_{Q}

ko integrate karne deta hai.

Is every uncountable set of positive measure?

Nahi — Cantor set uncountable hai phir bhi measure

0

hai.

What structure do the measurable sets form?

σ

-algebra (complement aur countable unions ke under closed), jisme saare Borel sets hain.

Recall Feynman: 12-saal ke bachche ko explain karo

Socho tum measure kar rahe ho ki points ke ek set ko kitni "ribbon" chahiye. Kisi bhi blob ko measure karne ke liye, tum uspar bahut saare chhote ribbon-pieces (intervals) daalte ho jab tak poora blob chhup na jaaye, phir unki lengths add karte ho. Tum har possible tarika try karte ho aur sabse kam total rakhte ho — yahi "size" hai. 0 se 1 tak ki normal stick ka size obviously 1 hai. Lekin yahan ek cool trick hai: fractions (1/2, 1/3, 1/4…) ko ek-ek karke list kiya ja sakta hai, toh tum $n$ -ve ko ribbon se cover karte ho jo pichle se aadhi chhoti hai. $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots$ add karo aur saare fractions milaakar bhi almost koi ribbon nahi chahiye — unka size $0$ hai! Kuch weird blobs itne messy hote hain ki unhe honestly measure nahi kiya ja sakta; hum sirf agree karte hain ki sirf well-behaved wale measure karenge.

Connections

Riemann integration — jise fix karne ke liye Lebesgue measure banaya gaya
Lebesgue integral — measurable functions aur is measure par built
Sigma-algebra — measurable sets ki structure
Borel sets — open intervals wala sabse chhota $σ$ -algebra; saare measurable
Axiom of Choice & Vitali set — non-measurable sets ka source
Cantor set — uncountable, measure zero
Outer regularity — measure ko open supersets se approximate karna
Countable vs uncountable — kyun $Q$ null hai lekin $[0, 1]$ nahi