|
%------------------------------------------------------------------------------
% Properties of sigma-finite measures
%
% Author: David Lester, Manchester University, NIA, Université Perpignan
%
% All references are to SK Berberian "Fundamentals of Real Analysis",
% Springer, 1991
%
% Version 1.0 1/5/07 Initial Version
%------------------------------------------------------------------------------
sigma_finite_measure_props[T:TYPE, (IMPORTING subset_algebra_def[T])
S:sigma_algebra, (IMPORTING measure_def[T,S])
mu:sigma_finite_measure]: THEORY
BEGIN
IMPORTING measure_contraction_props[T,S,mu],
measure_equality[T,S]
f: VAR nn_integrable[T,S,mu]
g: VAR integrable[T,S,mu]
h: VAR nn_measurable[T,S]
A: VAR (S)
x: VAR T
n: VAR nat
F: VAR sequence[[T->real]]
convergent?: MACRO pred[sequence[real]] = topological_convergence.convergent?
sfm_integrable: LEMMA
((FORALL n: integrable?[T,S,contraction(mu,A_of(mu)(n))](h)) AND
convergent?(series(lambda n:
integral[T,S,contraction(mu,A_of(mu)(n))](h))))
<=> integrable?[T,S,mu](h)
sfm_integral: LEMMA
convergence?(series(lambda n: integral[T,S,contraction(mu,A_of(mu)(n))](f)),
integral[T,S,mu](f))
sfm_component_eq: LEMMA x_eq(to_measure(fm_contraction(mu,A_of(mu)(n)))(A),
contraction(mu,A_of(mu)(n))(A))
IMPORTING integral_convergence[T,S,mu]
sfm_monotone_convergence: LEMMA % 7.4.5
ae_increasing?(F) AND
(FORALL n,x: F(n)(x) >= 0) AND
(FORALL n: integrable?[T,S,contraction(mu,P_of(mu)(n))](F(n))) =>
(((EXISTS g:
ae_convergence?(F,g)) <=>
bounded?(lambda n: integral[T,S,contraction(mu,P_of(mu)(n))](F(n))))
AND
FORALL g:
ae_convergence?(F,g) =>
converges_upto?(lambda n:
integral[T,S,contraction(mu,P_of(mu)(n))](F(n)), integral(g)))
END sigma_finite_measure_props
[ Dauer der Verarbeitung: 0.0 Sekunden
(vorverarbeitet)
]
|
