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Гомельский государственный университет им. Франциска Скорины

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[НЕСОРТИРОВАННОЕ]

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X P(X)

K P(X)

K =

A, B K A B K, A\B K

( , \) ( , ) (\, )

K = { } K = P(X)

K A, B A ∩ B K

A ∩ B = (A B)\(A B) = (A B)\((A\B) (B\A)) K

! A P(X) " # A A %

& K(A) K(A) ' # A
( ) * A #
K(A) = α Kα		(Kα A) K(A) ' K(A) = Kα, K(A),					A, B

K(A),	A, B	K		+	\	& '		K(A)
			α,	α A B Kα A B α Kα = K(A)

S P(X)

A, B S A ∩ B S		n
A B & A B A ∩ B =
, A, B S A B B\A = i=1 Ai Ai S Ai ∩ Aj =						i = j
	K K A K			= A\A
	S = {[a; b)\|a b} a, b R [a; a) = * , &
	S = {[a1, b1) × . . . × [an, bn)\| ai bi			i =		ai, bi R}
					1, n,		n
S K(S) ) S K(S) = i=1 Ai\| n N, Ai S
- & !
/ * n.			i				.
			n		p
S A S,			B = =1 Bi A\B = i=1 Ai Ai				S

n = 1 B = B1 A\B = A\B1 = A\(A ∩ B1)

k−1

n = k − 1

B = i=1 Bi,

A\ i=1 Bi = (A\ i=1 Bi)\Bk = 0 % , 1 = (

n(Ai )

S K(S)

= i=1 Ai\(Ai ∩ Bk ) = i=1( j=1 Aij ) mi = i=1 n(Ai) +

+ . ! &

K(S)

K(S) =

A, B K(S), A =

Ak , B =

Bk,

Ai, Bi S

Ai)\Bk = (Ai\Bk) =

i=1

			k=1	k=1	j
					j
		n		n		Akj \(	m
A B = (A\B) B = (			Ak \B) B = ( (Ak \B)) B =			Akj \(	Bk ) K(S)
		k=1		k=1	k=1,n		k=1

		k			=1,p		∞
n	m	n	p	Akj S 2 K(S)
A\B =	Ak \	Bk =		Akj S 2 K(S)
k=1	k=1	=1 j=1

( K σ A1, . . . An, . . . K							i=1 Ai K
( K δ A1, . . . An, . . . K							Ai K
							∞

i=1

A P(X) 3 " σ% # A σ A 4 & δ A

( K P(X) X K

5 R P(x) σ .

R σ%

5 R P(x) δ .

R δ%

P(X) σ% * δ% * / δ% σ% *

6 τ * &) -* + %

* σ% ) τ ' σ & ' B(τ )

S 7 µ : S → R S

µ(A		A
n	n
! A = k=1 Ak	µ(A) = k=1 µ(Ak ),	A1, . . . , Ak S
)			k=1 Ak	=
8 µ S ' σ% * + &) % * µ			k=1 Ak	=

			∞

∞

=µ(Ak )

k=1

µ S µ1 S1 µ1 ' µ

S1 .

S S1

µ1(A) = µ(A) A S

Ai S

µ1(A) :=

A K(S) +9 A =

Ak ,

(Ak ) A =

k=1

i=1

∩ Bi,

∩ Bi) S

Bi =

Cik

= Ak

Cik =

(Ak

Cik , µ(Bi) =

µ(Cik )

µ1

i=1

k=1

µ(Cik ) =

µ(Bi) =

µ(Cik ) =

µ(Ak ) = µ1

(A) µ1 *

i=1

i=1 k=1

k=1 i=1

k=1

+ &

(A)

= µ(A) A

S µ

µ K(S) ! & & µ

n(Ak )

Ak ,

Bki , Bki S.

K(S), A =

µ1(A) =

µ(Bkj ) =

µ1(Ak )

k=1

i=1

k=1

8 & µ σ% µ1 σ%

B µ(A)

µ(B)

A, B S : A n

B\A = k=1 Ak ,

Ai S,

B = A B\A = A (k=1 Ak ) µ(B) = µ(A) + µ(

=1 Ak )

µ(B) µ(A)

µ S C1 . . . Cn, . . . S &) C ∞ Ci µ(C) ∞ µ(Ci)

i=1

µ1 µ K(S) µ1(

Ci) µ1(C)

i=1

∞

i=1 µ1(Ci) µ1(C),

=1 µ1(Ci)

µ(C),

n N ' i=1 µ(Ci) µ(C)

n−1

µ S C1 . . . Cn, . . . S &) C i=1 Ci µ(C) =1 µ(Ci)

C = (C ∩ C1) ((C\C1) ∩ C2) . . . ((C\ Ci) ∩ Cn). µ(C) = µ1(C) = µ1(C ∩ C1) + µ1((C\C1) ∩ C2) +

∞

i ∞

+ · · ·

+ µ(C\ i=1 Ci) ∩ Cn) µ1(C1) + µ1(C2) + · · · + µ1(Cn) = i=1 µ1(Ci) = =1 µ(Ci)

µ σ% S C1, C2 . . . S : C =1 Ci

µ(C) i=1 µ(Ci)

. µ σ% S,

C = (C ∩ C1) ((C\C1) ∩ C2) . . . ((C\

=1 Ci) ∩ Cn)

µ(C) = µ1(C),

n−1

. . .

µ1 σ% µ(C) = µ1(C) = µ1(C ∩C1) + µ1((C\C1) ∩C2 ) + · · ·+ µ(C\

=1 Ci) ∩

∞

∩

Cn) . . . µ1(C1) + µ1(C2) + · · ·

+ µ1(Cn) = i=1 µ1(Ci) = i=1 µ(Ci ) +

=1 µ(Ci)

∞

&. & µ σ% A1, A2, . . . S

: A = i=1 Ai & &

=1 Ai

∞

µ(A) i=1 µ(Ai ) µ(A) i=1 µ(Ai) A i=1 Ai & µ(A) =

=1 µ(Ai)

#$ µ σ% S,

E1, E2, . . . S %

∞

{En} '# + E1 E2 . . . E =

∞

En µ(E) = µ(nlim µ(En)) = nlim µ(En)

n=1

→∞

E = E1 (E2\E1) (E3\E2) . . . µ(E) = µ(E1) = n=2 µ1(En\En−1) + µ(E1) = n=2(µ1(En) − µ1(En−1)) +

+ µ1(E1) =

A, B

K µ(B) = µ((B A)

A) = µ(B A) + µ(A)

lim

µ (E ) = µ(E)

µ(B\A) = µ(B) − µ(A)] = n→+∞

. . . En . . . ,

E =

µ σ% S {En}

∞

E1, E2, . . . S µ(E) = nlim (En)

=1 En,

→∞

∞

6 E1\E2 E1\E3 . . . E1\En . . . ,

E1 = n=2(E1\En) 6

K(S)

)

* :

µ1

K(S) +µ

(E1\E)

= lim

E ),

µ (E )

µ (E) = lim

(µ (E )

µ (E ))

µ (E) = lim

K(S) µ1

n→+∞

µ1(E1\ n

1 1

−

n→+∞

1 1

− 1

n→+∞ µ1(En).

6 S = {[a; b)| a, b R

a b} ,

µ ([a; b)) = b − a µ *

S '

! 8 < * σ% *

∞

[a; b) =

∞

[ak , bk)

[a; b) *

∞ µ([ak , bk)) µ([a; b)).

B = [a; b − 2ε ),

Bk = [ak −

; bk)

2k+1

k=1

[a; b − 2 )

[a; b)

[ak, bk )

Bk = B R

[ak ; bk)

k=1

(aα1 ; bα1 ), . . . , (aαn ; bαn ) : B =1(aαi ; bαi )

∞

B1, . . . , Bn : B

Bk µ([a; b −

2 ))

µ([ak −

; bk)) & [a; b − 2 )

[ak −

; bk )

2k+1

2n+1

k=1

∞

b − 2 − a

(bk − ak +

2k+1

b − a

(bk − ak) + 2 µ([a; b))

µ([ak ; bk )) + ε

k=1

ε → 0 + 0 & µ([a; b))

µ([ak , bk))

(

) " "

m % S % X A X

& &

A * +

∞

% A & inf m(Ak ) A Ak , Ak S / & A

k=1 k=1

& " ''

∞

& µ (A)

" * & & µ = +

" ) "

∞

µ : P(S) → [0; +∞] + '& +∞

∞

Ak , A X, Ak X k N µ (A) <

µ (Ak )

∞

k=1

>& & & *

µ(Ak ) < +∞

k=1

∞

ε > 0 " * k N Bki

S, i N : Ak i=1 Bki i=1 m(Bki ) < µ (Ak ) +

∞

µ (Ak ) = inf{

m(Ak )|A

Ak , Ak X}

k=1

∞

A k,i Bki µ (A) k,i=1 m(Bki )

k=1(µ (Ak ) +

) = k=1 µ (Ak ) + ε

, 8

A B, A, B X µ (A) µ (B)

∞

Bk % & &

& A B

? A

S µ (A) = m(A)k

A A m(A) % A S µ (A) m(A) A ∞ Ak % ? m(A) ∞ m(Ak )

m(A) µ (A) m(A) = µ (A)

k=1

2 * & " * S #

* P(X)

m % < R m([a; b)) = b − a 3 * µ ([0; 1])

[0; 1) [0; 1] [0; 1 + n1 ) µ ([0; 1)) µ ([0; 1]) µ ([0; 1 + n1 ))

n → ∞ & µ ([0; 1]) = 1

B % & & µ (B) = 0

& & b % & R µ (b) = 0 b [b, b+ n1 ) % µ(b) µ ([b, b+ + n1 )) = n1 µ (b) 0 µ (b) = 0

9 & µ (B) = 0 %

* , - (

8 % A X E X µ (E) = µ (A ∩ E) +

+µ (E ∩ (X\A)) (1)

> & < % M

µ(A) = µ (A) A M !+ %

M % σ%

m % σ% S µ @ * + σ% M +m %

" *

σ% + % M

! < % m! & & S M

		m % < * 9 [a, b) ' %
S	M 9 [a; b] =				∞ [a, b + 1 ) [a, b] % m([a, b]) = µ ([a, b])
					n=1	n
µ([a, b]) = lim µ([a, b +			1		% A		µ([a, b]) = b		a

			n	))				−
	n→∞
(a, b) = [a, b)\{a} µ((a, b)) = b − a
2 %
		-
[0; 1) "						x y x − y Q " [0; 1)

'# / ' %

! & M r + [x] := {r + x|x M } 6 & +% B &

∞

& & {rn} & [0; 1) n=1(rk + M ) [−1; 2] ( )

% µ(M ) = 0

µ(r + M ) = 0

k Q % ? 1

∞

−

µ(

(rk + M )) = 0 &

k=1

µ(M ) > 0

µ(M ) = µ(rk + M )

∞

rk + x = rl + y, rl = rk ' % , &

µ(rk + M ) = +∞

∞

µ(rk + M ) 3 & M %

k=1

( %

6 [0; 1] , & * ( 1 ;

2 ) *

" , & * & " % & F0

F0 = [0; 1]\G0 G0 % -

∞

2k−1

1 ∞

µ(G0) = k=1

2 k=1 3k = 1 µ(F0) = 1 − 1 = 0

& #

/ %

F0 & & * %

% * # D &

& &

. -($ $

S = {[a, b)|a b, a, b R} µf ([a, b)) = f (b) − f (a), f % '# R µf

f (x) = x < *

µf % σ% S f %

+ %

* /,

" /,

(X, Σ, µ) % % * Σ % σ% * σ% µ

7 f : E → R % E Σ E

{x E|f (x) < C} C R


& % E ∩ f −1((−∞, C))
7 ! D(x) = 1, x Q		' E
	0, x R\Q

C > 1 A = E %

C 0 A = %

0 < C 1 A = E ∩ (R\Q) = E\(E ∩ Q) % + E, Q %

1, x A

A % % XA(x) = '

0, x A

! * C = 12 {x E|XA < 12 } = {x E|XA = 0} = E\A A %

& E\A

3 @

{x E|f (x) < C} = E ∩ f −1((−∞, C)) % %

µ % < * &

7% g : R → R ' % B g−1(B) %

3 @ * &

g % f % g ◦ f

+ %

! 5 E @ * * 2

# & % @
f & kf, f + a, k, a = Const ! * C − a, Ck
' & C
> rk	x g(x) < rk - &
		∞

f g {x|f (x) > g(x)} % ! * {x|f (x) > g(x)} = {x|f (x) >

k=1

}∩{ | }

> ' & & {x|f (x) > a − g(x)} = {x|f (x) + g(x) > a}

, 1), 2)

? / f g = 4

((f + g)

− (f − g) ) 5 # %

@ % ,

: f (x) f (x) = 0

f (x) : & &

! % {fn(x)} : fn(x) % E n N

fn −→ f + nlim fn(x) = f (x) x E

f % E

→∞

fn(x) −→ f (x) {x|f (x) < C} =

{x|fm(x) < C − k1 } (1)

m n

! *

f (x) < C

* "

1 k : f (x) < C − k B k

m > n fm(x) < C − k & & x * ' & +

x * & + k & & " m

> 1

fm(x) < C − k f (x) < C

fn(x) {x|fm(x) < C − k } 9 % σ%

& + % % & f (x) %

8 µ S & E S, µ(E) = 0

F E µ(F ) = 0

< M * *

/ *" & & (X, Σ, µ) % * *

P (x) % * % = & % P (x) & ' + % x D

f = g µ({x|f (x) = g(x)}) = 0

7% E ' ! E

! E % f = g E f g

2 & {x|f (x) < C} {x|g(x) < C} & '

% D + *

4 +

! % E E f f % E

g(x) : fn(x) −→ g(x) g(x) = f (x) < & g #

fn f

9 g % 9 , f %

! 0 % E fn E @% f δ > 0 Eδ :

µ(Eδ ) > µ(E) − δ

f −→ f E

n −→ δ

= & % E @%* fn * @ f

δ > 0 lim µ({x E||fn(x) − f (x)| δ}) = 0

n→∞

! % E @%* E E

2 9 ? & f % A = {x E\| nlim fn(x) = f (x)} Ek (δ) = {x E\| \|fk (x) −
&		R1(δ)	→∞	µ(Rn(δ))
			R2(δ) . . .
∞	∞
− f (x)\| δ}, Rn(δ) = k=n Ek (δ), M = n=1 Rn(δ)					→
% 9

µ(M ) n → ∞

& M A x0 A δ > 0 n : |fk (x0) − f (x0)| < δ, k n

x0 Rn(δ) x0 M

3 µ(A) = 0 M A µ(M ) = 0 µ(Rn(δ)) → 0 n → ∞ 9 En(δ) Rn(δ)9

+6 * & + %

fnk = X[ k−n 1 ; nk ], k = 1, n, n = 1, 2, . . . % f11, f21, f22, . . .

! % {fn} E @%* f * E {fnk } : fnk −→ f

! 1 2 , 3 f % [a, b] ε > 0 @ ϕ : µ({x [a, b]|f (x) = ϕ(x)}) < < ε

9 * '

2 '

* (

( " /,

{X, Σ, µ}% *

7 ' h * E Σ h &

& & & *

h g h ± g hg % g(x) = 0, x E hg %

! % & & & & & &

		h	@%	{a1	, ..., an} % & * Ei = {x E\|h(x) = ai}
	n
	i
&	=1 aiµ(Ei) % + & = +∞ < @ h

& .

hdµ h(x)dµ(x)

& +∞

2 *" & &

+∞

(+∞) + (+∞) = (+∞) 0(+∞) = 0

* ( ($" /,

f % + x E, f (x) 0 9 < @% f & + & +∞

sup{ hdµ|h % h f E}

9/G & * @ %

+∞ & & +∞

" ( ($" /,

@% f g f dµ = gdµ (f, g %

f g E & % + & % f g & & D # < *

f (x), x E * D A = {x E|f (x) = g(x)} (µ(A) =

h % ' E : h(x)

= 0) @ * h & @ ' h1

: h1(x) g(x), x E / &

h(x) dµ =

h1(x) dµ % <

* @

, @%

@% f g

0 f (x)

g(x),

x E f dµ

gdµ

k R, k 0

E kf dµ = k E f dµ

ΣE = {F E|f Σ, . . F − }

% '

? >

F →

f dµ, f 0

A {fn} %

0 f1(x) f2(x) ... fn(x) ...

: f, g 0 (f + g)dµ =

f dµ + gdµ

@% * &

→∞

x E fn → f + 9

f dµ = nlim

fndµ

F +3 % H "

0, c > 0 µ{x E|f (x) c}

/ %

f dµ

A = {x E|f (x) C} f (x) CXA x E

% H "

f dµ =

0dµ = 0

I f

0 E f dµ = 0

f = 0

. .

∞ {x E|f (x) n1 }

f dµ = 0 A = {x E|f (x) = 0} =

n=1

µ(A)

∞

})

∞

µ({x E|f (x) n

(n f dµ) = 0

n=1

f dµ CXAdµ = Cµ(A) > '

J 4 ' < * @%.

+ %

δε > 0 : E1 E : µ(E1) < δε

f dµ < +∞ ε > 0

f dµ < ε

* (

min(f (x), 0)

f @% f +(x) = max(f (x), 0), f −(x) =

−

f = f + − f −, |f | = f + + f −

f dµ < +

f −dµ

< @%

% @%

7% f *

∞

f +dµ

− E

| |

f − |f |

5 * <.

f dµ

gdµ

f %

f g

f − g−

E f dµ = E f

dµ −E f −dµ E g

dµ −E g−dµ = E gdµ

| |

, f g 9 C R

Cf dµ = C

f dµ

(f + g)dµ = f dµ + gdµ

? 5& < .

∞

f E = En

E E E E E

En 9 f dµ =		f dµ &
	∞

n=1

: 4 ' < .

f ε > 0 δε > 0 : E1

E n=1 En

E µ(E1) < δε | f dµ| < ε

> 0

: E

E : µ(E

) <

(f +

− f

−) dµ

− dµ

( )

K & E f dµ =

E f + dµ

E f

& % @ * &

< δε

f + dµ

< 2ε

& ' *

2 δε

: E E : µ(E ) < δε f −

dµ

% +C

< * @

∩

A @%

∩ E

E : µ(E1) < δε

ε > 0

δε = min{δε

, δε } :

= E

E1 f dµ

< ε

f . .= 0 f + . . 0, f −

. . 0 + %

@ 5 f dµ =

f = 0

f dµ = 0

= f +dµ − f −dµ = 0 − 0 = 0 % @ *

( @%

9 f . . g f − g . . 0

(f − g)dµ = 0 > '

(f − g)dµ =

f dµ −

gdµ *

f dµ = gdµ

% @% @%

& & %

* @%

($" 4 (

5 , {fn}% @% * f 0

9 @%

lim

fn(x)

lim

fndµ lim fndµ

n→∞

= inf

, f

n+1

, . . .

}

{

{x E|gn(x) < C} =

{x E|fk(x) < C}

k n

	→∞
	lim	fndµ < +∞
	n	fndµ < +∞
		E

+L 2 & rlim gr = nlim

inf{gn, gn+1, . . .}

gn(x) gn+1(x) x E nlim fn = nlim gn = f (x)

→∞

& f & E A = {x E|f (x) = +∞}

∞

En = {x E|gn(x) C} & & En En+1 A

=1 En

n=1

n→∞

n C n→∞E

n C k n E

C n→∞

∞

µ(A)

µ(E ) =

lim µ(E )

lim

g dµ

inf

f dµ =

lim

C > 0 C → +∞ µ(A) = 0

* A {gn}

fn dµ

f dµ = lim gn dµ lim

n→∞E

n→∞

E 9 @% f

(

{fn}

n N

: K > 0 :

fndµ K n N

lim fn(x) = f (x), x

! " 4

% {fn} @% * & |fn(x)| ϕ(x) ϕ %

! 6-

{fn}

f dµ = nlim

fndµ =

nlim fndµ

fn −→ f 9

→∞E

→∞

@% * &

f1(x) f2(x) fn(x) ...

k R :

fndµ k , n N nlim fn(x) = f (x) f f dµ =

= lim

fndµ

→∞

1 ( 7

! @% 6 [a, b] < [a, b]

6 < '

6 [a, b] 2n & * & xk = a +

(b − a), k =

, x0 = a, x2n = b

0, 2n

−

sup

f (x)

, M

k k

x [xk−1,xk ]

k=1

−

inf

f (x)

k k

x [xk 1,xk ]

k=1

9 f 6 [a, b] nlim Sn

= nlim sn = I =

f (x)dx {

} {

} :

fn(x) = Mk , xk 1 x xk , k = 1, 2n

→∞

−

fn(x) = mk , xk−1 x xk , k = 1, 2n

/ & b @ D fn fn+1, x [a, b], n N

fn fn+1, x [a, b], n N

! &


% {		fn	} 9 = %< & &			nlim	fn	(x)dµ = nlim
% {			} 9 = %< & &		a,b]	nlim		(x)dµ = nlim
f (x) = lim fn(x)				[		→∞		→∞
	n→∞
	n→∞

fndµ = lim sn = I

[a,b]

n→∞

& {fn} & & nlim fn(x) = f (x) :

[

lim

(x)dµ =

lim

dµ = lim Sn = I

→∞

a,b]

[a,b]

[

< * @ f . .

(x) fn(x)

− f 0 x [a, b]

(

− f )dµ = I − I = 0 %

a,b]

[

9 fn f (x)

fndµ

f dµ

fndµ

a,b]

[a,b]

n → ∞ % &

! ( ( 7

! & @% 6 * <

& & * 6 '

< 6

+= %

8 ! 5,

Sx % M Sy % % Y Sx × Sy df {A × B|A X, B

Y } * % X × Y

Sx, Sy % Sx × Sy

B = B1 × B2, B1

A Sx × Sy , B Sx × Sy & & A = A1 × A2, A1 Sx, A2 Sy ,

Sx, B2 Sy 9 A ∩ B = (A1

∩ B1) × (A2

∩ B2) Sx × Sy ! \ &

Sy %

B1 A1, B2 A2

/ &

' .

A1 = B1 B1(1) . . . B1(k),

A2 = B2 B2(1) . . . B2(l),

A = A1 × A2 = (B1 × B2) (B1 × B2(1)) . . . (B1 × B2(l))

(B1(1) × B2) (B1(1) × B2(1)) . . . (B1(1) × B2(l))

. . . . . . . . .

(B1(k) × B2) (B1(k) × B2(1)) . . . (B1(k) × B2(l))

& B1 × B2 = B & Sx × Sy

µx % Sx µy % Sy 9 @ µx ×

×µy (A × B) = µx(A) × µy (B) A Sx, B Sy Sx, Sy

µx, µy % σ% µx × µy % σ%

X = R, Sx = {[a, b)|a b, a, b R}, Sy = {[c, d)|c d, c, d R}, µx, µy % <

* µx × µy ([a, b) × [c, d)) = (b − a)(d − c)

< µx × µy µx, µy µx µy

% A X × Y 8 % Ax0 = {y Y |(x0, y) A} x0% & % 4

! 5, µx, µy σ% @% f (x, y)
+ %			x			y
µ = µx µy % A X ×Y 9		f (x, y)dµ(x, y) = (	f (x, y)dµy (y))dµx (x) = (			f (x, y)dµx(x))dµy (y)
	A	X A		x	Y A		Ay
						Y
! ! ( # .				(	\|f (x, y)\|dµy (y))dµx(x)		( f (x, y)dµx(x))dµy (y)

X A

9 @% 4 7 + %

5& % ' @% ' Φ ' σ% Σ '# ' & &

	σ % Σ % σ%
	µ1, µ2 N σ% σ% Σ µ1 − µ2 *
	Φ(A) = f (x)dµ A X f @ X Φ %

Φ Σ +σ% % M 8 % A X

7 ' % B A : B Σ Φ(B) 0

4 & % +Φ(B) 0

! 7 M + σ% % M 9 # '

X+ X− : X = X+ X− A Σ : A ∩ X+ Σ, A ∩ X− Σ

+L ! & Φ % |Φ(A)| C A Σ C = Const A

a =

inf

>& &

{An}

nlim Φ(An) = a

−

Φ(A)

→∞

X : Φ(C0) < 0 C0

∞

% ! &

X+ = X\X−

X− = n=1 An >& X−

A = C

X−

Φ(A )

Φ(X

−) + Φ(C ) = a + Φ(C ) < a &

" & k1 C0

C1 C0(C1 = C0) : Φ(C1)

C0\C1

" & C2 : Φ(C2)

k2 > k1

X−

∞

F0 = C0\ i=1 Ci F0 = Φ(C0) < 0, Φ(Ci) > 0, i 1 2 & F0

& '

Φ(E) 0

6 X = X+ X− "

A Σ Φ+(A) := Φ(A ∩ X+) Φ−(A) := −Φ(A ∩ X−) >& & Φ+, Φ− %

Φ(A) = Φ+(A)

−

Φ−(A) + #

Φ(A) = Φ+(A) + Φ−(A)

Φ+, Φ− %

7 % * σ% % M µ % σ%

σ% % M K 7 ' % µ Φ(A) = = 0, A Σ & µ(A) = 0

! 27-# 3 µ % & σ% Σ + σ%
% M 7 M ' % * µ 9 ! & '
* 7 & ' .dΦ
@% f ' % A Σ : Φ(A) =	f dµ @% f

dµ

+ %

7 % * M+= 0 ' * µ

9 n N % B & / % % Φ − n1 µ + %

: 5, ; ( /,

: 5, "

f (x) [a, b]

" 4 /, "

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