%This command provides the two last columns of
%the second horizontal part of the page 10
%
%The macro has one parameter:
%       1) The width available to typeset the formulae
\newcommand\TTenStieltjes[1]{%
   %This command print one line of the array showing
   %the differents Stieltjes integrals. 
   %
   %The command has two parameters:
   %      1) The part of equation before the equal sign
   %      2) The part of the equation following it
   \def\LineOfArray##1##2{%
        ##1&=&##2\\[\TTenExpansionSkip]%
   }
   \parbox[t]{#1}{%
      \TTenStieljesFontSize
      \DisplaySpace{\TTenDisplaySpace}{\TTenDisplayShortSpace}
      \noindent
      If $G$ is continuous in the interval $[a,b]$ and $F$ is nondecreasing then
      \begin{displaymath}
         \int_a^b G(x) \, d F(x)
      \end{displaymath}
      exists.

      \AdjustSpace{1.5ex plus .5ex minus 1ex}
      If $a \leq b \leq c$ then
      \begin{displaymath}
         \int_a^c G(x) \, d F(x) = \int_a^b G(x) \, d F(x) + \int_b^c G(x) \, d F(x)
      \end{displaymath}

      \AdjustSpace{1.5ex plus .5ex minus 1ex}
      If the integrals involved exist
      \begin{displaymath}
         \begin{array}{lcl}
            \LineOfArray{\int_a^b \big(G(x) + H(x)\big)\, d F(x)}%
                        {\int_a^b G(x) \, d F(x) + \int_a^b H(x) \, d F(x)}%
            \LineOfArray{\int_a^b G(x)\, d \big(F(x) + H(x)\big)}%
                        {\int_a^b G(x) \, d F(x) + \int_a^b G(x) \, d H(x)}% 
            \LineOfArray{\int_a^b c \cdot G(x)\, d F(x)}%
                        {\int_a^b G(x)\, d \big(c \cdot F(x)\big) = c \int_a^b G(x) \, d F(x)}%
            \LineOfArray{\int_a^b G(x)\, d F(x)}%
                        {G(b)F(b) - G(a)F(a) - \int_a^b F(x) \, d G(x)}%
         \end{array}
      \end{displaymath}
      If the integrals involved exist,
      and $F$ possesses a derivative $F'$ at every point in $[a,b]$ then
      \begin{displaymath}
         \int_a^b G(x) \, d F(x) = \int_a^b G(x) F'(x) \, dx
      \end{displaymath}
   }%
}
%The title of this part
\newcommand\TTenTitleStieltjes{Stieltjes Integration}

.
