SSL fubini_tonelli_scaf.pvs Sprache: PVS

%------------------------------------------------------------------------------
% Fubini-Tonelli Theorems
%
%     Author: David Lester, Manchester University, NIA, Universite Perpignan
%
% All references are to SK Berberian "Fundamentals of Real Analysis",
% Springer, 1991
%
%     Version 1.0            5/06/09  Initial Version
%------------------------------------------------------------------------------

fubini_tonelli_scaf[(IMPORTING subset_algebra_def, measure_def)
       T1,T2: TYPE,
       S1:sigma_algebra[T1],S2:sigma_algebra[T2],
       mu:sigma_finite_measure[T1,S1],
       nu:sigma_finite_measure[T2,S2]]: THEORY

BEGIN

  IMPORTING product_measure[T1,T2,S1,S2],
            integral[[T1,T2],sigma_times(S1,S2),m_times(mu,nu)]

  E:   VAR (sigma_times(S1,S2))
  X:   VAR (S1)
  Y:   VAR (S2)
  x:   VAR T1
  y:   VAR T2
  i,j,n: VAR nat

  IMPORTING product_integral_def[T1,T2,S1,S2,mu,nu],
            measure_contraction_props,
            measure_equality,             % Proof only
            finite_fubini_tonelli,        % Proof Only
            finite_fubini,                % Proof Only
            indefinite_integral,          % Proof only
            extended_nnreal@double_nn_sequence, % Proof only
            sigma_finite_measure_props    % Proof Only

  h: VAR nn_measurable[[T1,T2],sigma_times(S1,S2)]

  IMPORTING product_integral_def,      % Proof only
            sigma_finite_measure_props % Proof only

  convergent?: MACRO pred[sequence[real]] = topological_convergence.convergent?

  product_measure_contraction: LEMMA                                % SKB 7.4.6
    x_eq(contraction(m_times(mu, nu),cross_product(X,Y))(E),
         m_times(contraction(mu,X),contraction(nu,Y))(E))

  product_sfm_contraction: LEMMA
    x_eq(contraction(m_times(mu, nu),
                     cross_product(A_of(mu)(i),A_of(nu)(j)))(E),
         product_measure_approx(mu,nu)(i,j)(E))

  product_measure_contraction_n: LEMMA
    x_eq(m_times(contraction(mu, P_of(mu)(n)),
                 contraction(nu, P_of(nu)(n)))(E),
         to_measure(fm_times(fm_contraction(mu, P_of(mu)(n)),
                             fm_contraction(nu, P_of(nu)(n))))(E))

  fubini_tonelli_scaf1: LEMMA                                           % 7.4.7
    (integrable?(h) <=> integrable1?(h)) AND
    (integrable?(h) => integral(h) = integral(integral1[T1,T2,S1,S2,mu,nu](h)))

  fubini_tonelli_scaf2: LEMMA                                           % 7.4.7
    (integrable?(h) <=> integrable2?(h)) AND
    (integrable?(h) => integral(h) = integral(integral2[T1,T2,S1,S2,mu,nu](h)))

END fubini_tonelli_scaf

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