realanalysis08s-Lebesgue测度

实变函数课件

LebesgueVersionβ1.201, 01, 02, 01

2012 4 18

实变函数课件

2/14

1 S, (1) ∈ S; (2) E, F∈ S, E∩F∈ S; (3) E, F∈ S, S1,···, Sm∈ S, E F= m Sk . k=1 S . 1 R1 Rn ., P={(a, b] a, b∈ R, a≤ b} . n, P={ i=1 (ai, bi] ai, bi∈ R, ai≤ bi}

实变函数课件

3/14

1( E (1) (2) 1 S

), E E, F∈ E, E F∈ E; E, F∈ E E∩ F= ,

: E∪F∈ E.

R(S )={ 2 Rn

m

, Sk Sk∈ S, m∈ N}.

k=1

, R0:= R(P)

.

实变函数课件

4/14

2 S, m: S→ R S1, S2,···, Sk, T1, T2,···, Tl∈ S, S1, S2,···, Sk, T1, T2,···, Tlk l k l

.

.i=1

Si=j=1

Tj,i=1

m(Si )=j=1

m(Ti ).

m: S→ R m: R(S )→ R , S1, S2,···, Sk∈ S . m m R(S ) m(

,k k

Si )=i=1

m(Si ),

i=1

.

实变函数课件

m3 S . (1) m R(S ) (2) R(S ) E∈ R(S ) (3)k k i=1

5/14

,

m: S→ R m: R(S )→ R . E1, E2,···, Ekk

:,

Ei E,i=1

m(Ei )≤ m(E).

,m R(S ) . E1, E2,···, Ek∈ R(S ), E∈ R(S )k

E i=1

Ei,

m(E)≤i=1

m(Ei ).

实变函数课件

m2 S, m: S→ R (1) m( )= 0; (2) E∈ S, m(E)≥ 0; (3) S{Ek},∞

6/14

Ek∈ S,

∞

∞

m(k=1

Ek )= R(S )k=1

m(Ek ). . m:P→R

m: R(S )→ R n

k=1

Rn (ai, bi])= P

n i=1

P. (bi ai ). .

m(

m( )= 0, m

i=1

实变函数课件

LebesgueRn P. m:P→R mσ R0 L:= R0 R0 n n

7/14

m(

(ai, bi])=,

(bi ai ).i=1

i=1

R0= R(P) m Lebesgue m Lebesgue L,

m . . .

m. .

Lebesgue

m . Lebesgue

(Rn, L, m)

实变函数课件

Borel3 (1) Borel (2) Borel (3) (4) (5) . (6)

LebesgueB= Rσ (R0 ) Lebesgue Lebesgue Lebesgue Lebesgue R1 . 0. .σ

8/14

.

F . F (a, b), m(F )= (b a) m((a, b) F )., Cantor Lebesgue 0., m(F )= lim m(F∩[ c, c]).c→+∞

实变函数课件

9/14

4 (1) H (R0 )= 2R . (2) m (E)= inf{m(G) E G, G∈ T}. 4( )

A, B, C Rn . (1) m (A)= 0, m (A∪ B)= m (B). (2) m (A), m (B)<+∞,|m (A) m (B)|≤ m (A B) (3) m (A B)= 0, m (B C)= 0, m (A C)= 0.

实变函数课件

Lebesgue3 (Lebesgue )

10/14

E R1 .: (1) E Lebesgue . (2) ε> 0, G E, s.t. m (G E)<ε. (3) ε> 0, F E, s.t. m (E F )<ε. (4) Gδ H E, s.t. m (H E)= 0. (5) Fσ K E, s.t. m (E K)= 0. (6) Gδ H E Fσ K E, s.t. m (H K)= 0.

实变函数课件

11/14

A B m (B A)= 0, m (A)= m (B). . H E, E R, Lebesgue m (H)= m (E), . H E

1 (1) E R1, E . (2) Lebesgue, Fσ Gδ Gδ . H, H

实变函数课件

12/14

4 Borel 5( Lebesgue ) E, .

E R1 . m (E)> 0, Lebesgue .

实变函数课件

Lebesgue - Stieltjes

13/14

R . R mg: P→ R······

Lebesgue - Stieltjes g. (a, b]= g(b) g(a).

实变函数课件

14/14

: p144, 1, 2, 4, 5, 9, 11, 16, 18.