fourier jr
Nov3-04, 09:28 PM
problem: Let f \in BV[a,b] . Then \int_{a}^{b} |f'| \leq T_a^b f where T_a^b f is the total variation of f over [a,b].
there are some lemmas, etc that got me this far:
\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 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:
\vert \int_E f \vert \leq \int_E |f|
would that help at all?
there are some lemmas, etc that got me this far:
\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 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:
\vert \int_E f \vert \leq \int_E |f|
would that help at all?