Suppose f is integrable on [a,b], show |f| is integrable on [a,b];
I know I have to show Sp|f|-sp|f|<=Spf-spf
But I'm stuck from there
The usual way goes something like this.
Define two new functions: $\displaystyle F^ + (x) = \left\{{\begin{array}{rl}
{F(x),} & {0 \leqslant F(x)} \\ {0,} & {F(x) < 0} \\ \end{array} } \right.\;\& \,F^ - (x) = \left\{ {\begin{array}{r,f} { - F(x),} & {F(x) < 0} \\ {0,} & {F(x) \geqslant 0} \\ \end{array} } \right.$
We see that $\displaystyle |F(x)|=F^+(x)+F^-(x)$ so show each of those is integrable.