Fubini's Theorem

Fubini's Theorem :-

      In the world of mathematics integration plays an important role . So we have discussed the advance form of integration which results the famous theorem Fubini's Theorem.

Statement:

             If a double integral , I = ∫∫ f dx dy 

                                                       R

  exists over a rectangle R = [a,b;c,d] , and if 

   d

   ∫ f dy also exists , for each fixed x in [a,b] , 

  c                                              b      d

   then the iterated integral ∫ dx ∫ f dy exists

                                                a       c

   and is equal to double integral I.

Proof :

      Let ε  be any positive number.

  Since the upper integral , Iᵘ is the infimum of the upper sums , there exists a partition P of R such that 

               Σ Σ Mᵢⱼ ΔRᵢⱼ < Iᵘ + ε

                i  j

   or         Σ Σ Mᵢⱼ Δxᵢ Δyⱼ < Iᵘ + ε    ...........(1)

                 i  j

    Again , since for any fixed value of x ∈ Δxᵢ
, Mᵢⱼ is upper bound (not necessarily the supremum) of f in Δyⱼ , therefore
                   _d
                  ∫ f dy ≤ Σ Mᵢⱼ Δyⱼ , when x∈ Δxᵢ
                  c             j                       ..........(2)

     Again , from equation(2) , Σ Μᵢⱼ Δyⱼ is upper bound of the function
                            d
               Φ(x) = ∫ f dy , x∈ Δxᵢ
                           c

therefore by the same reasoning as above , we get
                    b
                  ∫̅  Φ(x) dx ≤ Σ(Σ Μᵢⱼ Δyⱼ)Δxᵢ
                   a                   i   j

 or
            _b    _d
           ∫ dx ∫ f dy ≤ Σ Σ Mᵢⱼ Δyⱼ Δxᵢ < Iᵘ +ε
           a       c             i  j  [by(1)]    ........(3)

    Also , since by hypothesis ,
                 d           _d           d
                ∫ f dy = ∫ f dy = ∫ f dy
                ̅ c           c             c

and ε is an arbitrary positive number , we get from equation(3),
              _b     _d          _b     d
              ∫ dx ∫ f dy = ∫ dx ∫ f dy ≤ Iᵘ ........(4)
              a      c             a       c

  By considering the lower integral Iₗ, we can similarly show that
                b     _d
              ∫ dx ∫ f dy ≥ Iₗ              ..........(5)
              ̅ a     a

 Again , since Iₗ = Iᵘ = I as the double integral exists , from (4) and (5) , we get
             b      d           _d
      I ≤ ∫ dx ∫ f dy ≤ ∫ f dy ≤ I
           ̅a       c            a
        b      d           _b     d
⇒   ∫ dx ∫ f dy = ∫ dx ∫ f dy = ∫∫ f dx dy
      ̅ a      c            a       c              R
            b  d
 Thus ∫ [∫ f dy ] dx exists and equals
          a   c 

      ∫∫ f dx dy .                                   (Proved)
       R 

Note :

             

                                                     b

1.  Similarly , if ∫∫ f dx dy and ∫ f dx both exist

                             R                     a

     then the double integral can be expressed as

                                      d      b

                ∫∫ f dx dy = ∫ dy ∫ f dx 

                 R                  c       a

2.   Since ε is an arbitrary positive number , inequality (3) shows that

        _b      _d             __

        ∫ dx  ∫ f dy   ≤  ∫∫  f dx dy 

         a      c                 R

      and similarly by considering the lower integrals ,

            b      d

           ∫ dx ∫ f dy  ≥ ∫∫ f dx dy 

           ̅ a      ̅ c            ̅ ̅ ̅ R

Remarks :

 1.  The theorem holds even if f has a finite        number of discontinuities , or an infinite      of discontinuities lying on a finite                    number of lines x = Cᵢ , i= 1,2,...m parallel      to y-axis .

                                                    d

       For , the function Φ(x) = ∫ f dy will be 

                                                   c

    discontinuous at a finite number of                points C₁,C₂,.........,Cₘ only and will                    therefore be still integrable . A similar            remark holds for discontinuities parallel      to x - axis .

2.   If a double integral exists , then the two        repeated integrals cannot exist without          being equal . However , if the double              integral does not exist , nothing can be          said about the repeated integrals ; they          may or may not exist . 

        Also one of the repeated integrals may        exist or even that both may exist and be        equal and yet the double integral may            not exist , i.e. the existence of one or of          both of the repeated integrals is no                  guarantee for the existence of the                    double integrals .

        However , if the two repeated integrals        exist both are unequal , the double                  integral cannot exist . 

Example :

          A function is defined on a rectangle [0,1;0,1] as 

                         { 1/2 when y is rational

           f(x,y) = {

                         { x ,   when y is irrational

   Prove that the iterated integral  

   1      1

  ∫ dy ∫ f dx = 1/2 , but the other iterated

  0      0

 integral does not exist . 

Proof :

                                  1           1

    For y rational    ∫ f dx = ∫ 1/2 dx = 1/2

                                0             0

    For y irrational 

                       1            1

                      ∫ f dx = ∫ x dx = 1/2

                      0            0

                1       1           1

∴             ∫ dy ∫ f dx = ∫ 1/2 dx = 1/2 

               0       0            0

     Proceeding as for a function of a single 

variable , we may show that the integral 

  1                                                1      1

 ∫ f dy does not exist and so ∫ dx ∫ f dy 

 0                                                0       0

also does not exist .

       It was shown earlier that the double integral ∫∫ f dx dy does not exist .

                  R

                                                          (Proved)

About Scientist :-

       Guido Fubini (19 January 1879 – 6 June 1943) was an Italian mathematician, known for Fubini's theorem and the Fubini–Study metric. Born in Venice, he was steered towards mathematics at an early age by his teachers and his father, who was himself a teacher of mathematics.For more

Recommended Books : 

Measure And Integration by Springer for undergraduate studies.

Concise Introduction To Measure Theory for undergraduate students .