As we know,if $f(x)$ are Riemann integrable,we have
\begin{gather} \left|\int_a^b f(x)~\mathrm{d}x\right|\leq \int_a^b |f(x)|~\mathrm{d}x. \end{gather}
So,for Darboux integral,such as upper integral,did we also have
\begin{gather} \left|\bar{\int_a^b} f(x)~\mathrm{d}x\right|\leq \bar{\int_a^b} |f(x)|~\mathrm{d}x\tag{1} \end{gather} right?
But how to prove it?
I think,we could use Darboux theorem.
Let $P$ is a partition (segmentation) on closed interval $[a,b]$. And denote $\lVert P\rVert$ is largest of the lengths of the intervals of the partition P.We have
\begin{gather} \lim_{\lVert P\rVert\to 0}\sum_{i=1}^n M_i\Delta x_i=\bar{\int_a^b} f(x)~\mathrm{d}x, \end{gather} $M_i$ is the supremum on $[x_{i-1},x_i].$ Then we use Darboux sum have,
\begin{gather} \left|\sum_{i=1}^n M_i\Delta x_i\right|\leq \sum_{i=1}^n |M_i|\Delta x_i\leq \sum_{i=1}^n \sup_{x\in \Delta x_i}|f(x)|\Delta x_i.\tag{2} \end{gather} Then I try to make limit on left hand and right hand,but this limit is not function's limit,they have different definition.So I decide to prove these limits'inequalities.
Firstly,I give Thm1:For two Darboux upper sum S(f;P) and S(g;P), P is a partition on [a,b],if \begin{gather} \lim_{\lVert P\rVert\to 0}S(f;P)<\lim_{\lVert P\rVert\to 0}S(g;P). \end{gather} Then,exist a $\delta >0$ such that \begin{gather} S(f;P)<S(g;P) \end{gather} forall $P$ and point $\xi$ with $\lVert P\rVert<\delta.$
Proof :Let \begin{gather} A:=\lim_{\lVert P\rVert\to 0}S(f;P),\quad B:=\lim_{\lVert P\rVert\to 0}S(g;P). \end{gather} Then we can find $C\in \mathbb{R}$ with $A<C<B.$So exist $\delta>0$,forall partition P and points $\xi$ with $\lVert P\rVert<\delta$,we have
\begin{gather} |S(f;P)-A|<C-A,\quad |S(g;P)-B|<B-C. \end{gather} Then we have \begin{gather} S(f;P)<A+(C-A)=C=B-(B-C)<S(g;P). \end{gather} Obviously we have $S(f;P)<S(g;P).$
Next,I give Thm2:For two Darboux upper sum S(f;P) and S(g;P),if exist $\delta >0$ such that forall P and $\xi$ with $\lVert P\rVert <\delta$,will have \begin{gather} S(f;P)<S(g;P). \end{gather} then,we have \begin{gather} \lim_{\lVert P\rVert\to 0}S(f;P)\leq\lim_{\lVert P\rVert\to 0}S(g;P). \end{gather} Proof:If \begin{gather} \lim_{\lVert P\rVert\to 0}S(f;P)>\lim_{\lVert P\rVert\to 0}S(g;P). \end{gather} It means exist $\delta >0$ such that for any P with $\lVert P\rVert<\delta$ have \begin{gather} S(f;P)>S(g;P). \end{gather} But it is a contradiction.We finished proof.
I'm not sure these theorems whether correct and rigor or not.If they are right,could I use it on inequality (2) to prove (1)?