Wp/lus/Tehhmunzirna

En bawk la : Hmunramzirna, Khawnkhawmzirna

Tehna

Tehhmunzirna hi chhiarkawp péng pakhat a ni a, tùnlai chhiarkawp sang péng tam zâwk hi tehhmunzirna-a innghat a ni. Suihkhawmna thuvàwr (integration theory) pumpui hi tehhmunzirna hmang lo chuan a nepnawi ho të të chin bâk a zir theih loh. Suihkhawmna thuvàwr leh tehhmunzirna hi a ruala zir ṭhin a ni, a bulṭanna pawl hniam lamah tih loh chuan.

Thil mumal taka tehna chungchang zirna a ni deuh ber a, entirnan, khaw inkar hlatzawng tehna te, huan zauzawng tehna te, tuithawl tui dawn hlâwkzawng tehna te leh thil dang, tehfung hmanga teh theih chi zawn zawngte tehna, chhiarkawp dàn mumal tak hmanga zirna a ni. Hetia kan hmuh theih khawvêl bâkah hian chhiarkawp mit chauhva hmuh theih thil dang tam tak tehna zirna a ni.

A lo inṭan dàn

Tehhmunzirna hi kum zabi 19-na tawp lam leh kum zabi 20-na tir lama Emile Borel-a te, Henri Lebesgue-a te, Radon-a te duan chhuah a ni a, mümal taka duang chhuaktu hmasa ber chu Henri Lebesgue-a a nih avangin Lebesgue-a thuvàwr (Lebesgue theory) ti pawhin an sawi bawk ṭhin.

A chhiarkawp

$X$ hi khawnkhâwm lo ni ta se, $\mathfrak{M}$ hi $X$ khawnpéng chhungkua lo ni ta bawk se la, $\mathfrak{M}$ hian a hnuaia nihphungte hi a neih chuan Tehhmunramthla tiin kan ko:

1. $\mathfrak{M}$ hi a ruak lo. Chumi awmzia chu, chhungkhung pakhat tal a nei.
2. $A$ hi $\mathfrak{M}$ chhunga a awm chuan, $A$ bâk (chu chu$X\setminus A$ tihna) pawh $\mathfrak{M}$-ah a awm ve.
3. $A_1, A_2, ...$ te hi $\mathfrak{M}$-a a awm chuan, a suihkhâwm

   $\bigcup_{n=1}^\infty A_n=A_1\cup A_2\cup\cdots$

   pawh hi $\mathfrak{M}$-ah a awm ve.

Hetiang nihphung nei $\mathfrak{M}$ hi kan neih chuan kawpchawi $\left( X,\mathfrak{M}\right)$ hi Tehnahmunram tiin kan ko va. Hnathawh

$\mu\colon\mathfrak{M}\to \mathbb{R}$

a hnuaia nihphung nei hi kan neih chuan $\mu$ hi tehna emaw tehkhäwng emaw tehfung tiin kan ko:

1. Dinna (= dinglam nihna/positivity) : $\mu (A)\geq 0\forall A\in \mathfrak{M}$
2. Chhiartheihbelhna (countable additivity): $\lbrace A_n\rbrace_n^{\infty}$ te hi $\mathfrak{M}$ chhungkhung, a tawn tawna intawklawi lo ni ta se,

   $\mu (\bigcup_{n=1}^{\infty} A_n)=\sum_{n=1}^\infty A_n$

   tih hi kan nei.
3. Khawnkhâwm ruak tehzawng bial: $\mu (\empty)=0.$

Hetiang hnathawh $\mu$ hi kan neih chuan thumchawi $(X, \mathfrak{M},\mu )$ hi Tehhmun tiin kan sawi a, hnathawh $\mu$ hi, kan sawi tawh angin, tehfung emaw, tehna emaw, tehkhäwng tiin kan sawi ṭhin. A tehhmunramthla $\mathfrak{M}$ kan neih sa emaw, hriat sâ anga kan ngaih chuan $X$ hi Tehhmun angin kan sawi bawk, a thumchawi $(X,\mathfrak{M},\mu )$ ziah buai kher ngai lovin.

Hetiang thil pathumte: khawnkhawm $X$, tehhmunramthla $\mathfrak{M}$ leh tehfung $\mu$ te hi kan neih kim sa vek chuan, "$X$ hi tehhmun lo ni ta se" tih thü te hi a sawi theih tawh ang.

Tichuan thil pathumte khi lo nei ta ila, $A\in\mathfrak{M}$ lo awm bawk se, $\mu(A)$ hi "$A$ tehzawng" tiin kan sawi ṭhin. Kan tehhmun hi thil zauzawng tehna lam hawi a nih chuan tehzawng tih aiah zauzawng tiin kan thlâk a, thil dawnhlâwkzawng tehna lam hawi a nih erawh chuan dawnhlâwkzawng tiin, a remchan dan ang zëlin kan thlâk. Amaherawh chu, avai huap chuan tehzawng kan ti mai thung.

Entirna

1. $X$ khawnkhawm ruak lo engpawh lo ni se, $\mathfrak{M}$ hi $X$ khawnpéng zawng zawng chhungkua lo ni ta se, hnathawh $\mu$ hi 0 veka kan dah

   $\mu(A):=0\forall A\in\mathfrak{M}$

   chuan tehhmun $(X,\mathfrak{M},\mu )$ kan nei.
2. $\mathbb{R}$-ah hian tehna pangngai, seizawng tehna, Lebesgue-a tehna tih chu kan nei. Hemi nihphung pawimawh tak pakhat chu, $\left[ a,b\right]$ tehzawng chu a seizawng pangngai, $b-a$ hi a ni. Entirnan, Lebesgue-a tehfung hmanga 1 leh 3 inkar hlatzawng kan teh chuan, $3-1=2$ kan hmuchhuak.
3. $X=\lbrace a, b, c\rbrace$ lo ni ta se, $\mathfrak{M}$ hi $X$ khawnpéng zawng zawng chhungkua lo ni ta bawk sela. $A$ hi $X$ khawnpéng engpawh lo ni se,

   $\mu(A):=A$ chhungkhung neih zât, tiin lo dah ta ila, $(X,\mathfrak{M},\mu )$ hi tehhmun a ni.

A pawimawhna

Atira kan sawi tawh angin, tehhmunzirna tel lo chuan Suihkhawmna thuvàwr hi a awm thei lo va, chu mai bâkah ama pual hian chhiarkawp ṭhang duang tak leh zau tak a ni. A hmanna dang ṭhenkhat:

• Harmonic Analysis-ah
• Ergodic Theory-ah

Thulâkna

• Rudin, Walter, Real and Complex Analysis, 3rd edition, McGraw-Hill Inc., 1987.
• Hewitt, Edwin & K.A. Ross, Abstract Harmonic Analysis Volume I, Structure of Topological Groups, Integration Theory, Group Representations, Springer-Verlag, Die Grundlehren der Mathematische Wissenschaften, Band 115, 1963.