19 Lebesgue Riemann

19 Lebesgue

19 Lebesgue LeviE {fn} limEfn(x)dx =Elimfn(x)dx (1), {fn}Fatou Elimfn(x)dx limEfn(x)dx (2)

19 Lebesgue

1 { fn }

19 Lebesgue

19 Lebesgue {fn}(1)

2{fn} (1)

19 Lebesgue ELebesgue{ fn} ffELebesgue1

19 Lebesgue { fn }ELebesgue f>0 Nn>N |fn(x)-f(x)|< (xE) |fn(x)| < |f(x)|+ (xE) 3

19 Lebesgue E|f(x)|+ E3

Lebesgue Levi,{ } ,f:

19 Lebesgue 2 0 x(1/n,1) fn(x) = n x (0,1/n] (1)?

19 Lebesgue fn(x) 010 n 01 1 E{ fn}

19 Lebesgue {fn}E fEI {fn}EfIIELebesgueF n |fn(x)| F(x) a.e.[E]

19 Lebesgue

3III { fn } 1

19 Lebesgue E fnfEgoroff >0 EE am(E- E)< ; bfnEf lim E fn (x)dx = E limfn(x)dx (3)

19 Lebesgue E- E Riemannf[a,b], >0,>0[c,d][a,b]d-c<

19 Lebesgue E- E { fn } F E- E |fn(x)| dx E- EF(x)dx4 Riemann

19 Lebesgue 1Lebesgue

19 Lebesgue Ef(x)EL-|f(x)|EL-f(x)E f(x)

19 Lebesgue

19 Lebesgue

4f(x)E f(x)

19 Lebesgue

f(x)

19 Lebesgue N

EA

19 Lebesgue

f(x)

19 Lebesgue , :(Lebesgue){ }E FEL - (1). a.e.[E](2). a.e.[E]fE