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Total variation problem

  1. Nov 3, 2004 #1
    problem: Let [tex]f \in BV[a,b] [/tex]. Then [tex]\int_{a}^{b} |f'| \leq T_a^b f [/tex] where [tex]T_a^b f [/tex] is the total variation of f over [a,b].

    there are some lemmas, etc that got me this far:
    [tex]\int_{a}^{b} f' \leq f(b)-f(a) = P_a^b - N_a^b \leq P_a^b + N_a^b = T_a^b f [/tex] where P is the positive variation & N is the negative variation of f.

    the absolute value there messes me up; i don't know what to do about it. i know there's a theorem that says the following:
    [tex]\vert \int_E f \vert \leq \int_E |f| [/tex]
    would that help at all?
  2. jcsd
  3. Nov 4, 2004 #2
    up... can anyone help?
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