Difference between upper integral and lower integral I have the following problem:

Let $f: [a,b]\longrightarrow \mathbb{R}$ a bounded function. Prove that
  $$\bar\int_a^b f-  \underline\int_a^b f=\bar\int_a^b w(f,x)dx$$

where $w(f, x)$ is the oscillation of $f$ on $x$. For a definition see here http://en.wikipedia.org/wiki/Oscillation_(mathematics)#Oscillation_of_a_function_at_a_point
I want to know if the following is correct:
Let $\epsilon>0$. For each $x$ there exist an open interval $I_x$ such that
$$w(f,I_x)-w(f,x)<\displaystyle\frac{\epsilon}{2(b-a)}$$
The family containing $I_x$ is an open cover of $[a,b]$. Since $[a,b]$ is compact there exist a finite subcover $I_{x_1},..., I_{x_n}$ wich cover $[a,b]$. Let $P$ a partition consisting on border points of $I_{x_i}$ that lie in $[a,b]$ and the points $a,b$. For this partition we have
$$w(f,I_{x_i})-M_i(w)\leq\displaystyle\frac{\epsilon}{2(b-a)}<\displaystyle\frac{\epsilon}{(b-a)}$$
then taking the sum we get
$$U(f,P)-L(f,P)-U(w,P)<\epsilon$$
since $\epsilon$ is arbitrary we have the result. 
Is this fine?
Thanks a lot!!!!
 A: The theorem appears (for the first time ?) in de la Vallée-Poussin Cours d'Analyse Infinitésimale Tome I (1903), pp. 218-219. It contains some wrong arguments as was shown by S.Pollard in Note on a Proof in Vallée-Poussin's "Cours d'Analyse", Messenger of Math. Vol. 44 (1914), pp. 141-144.
I refer to the proof (similar to yours) contained in E.W.Hobson The Theory of Functions of a Real Variable and the Theory of Fourier's Series Vol. 1 (1921), pp. 437-438. Till now I didn't find a more recent version.
Now your proof and my notes (in italics) avoiding some detail:
Let $\epsilon>0$ and $\delta>0$. For each $x$ there exist an open interval $I_x$ of length less than $\delta$ such that
$$w(f,I_x)-w(f,x)<\displaystyle\frac{\epsilon}{b-a }$$
The family containing $I_x$ is an open cover of $[a,b]$. Since $[a,b]$ is compact there exist a finite subcover $I_{x_1},..., I_{x_n}$ wich cover $[a,b]$.$\;$ (the existence of $P$ must be proved, see Hobson) Let $P$ a partition consisting of border points of $I_{x_i}$ that lie in $[a,b]$ and the points $a,b$. For this partition we have
$$0\le w(f,I_{x_i})-M_i(w)<\displaystyle\frac{\epsilon}{(b-a)}$$
then taking the sum we get
$$0 \le U(f,P)-L(f,P)-U(w,P)<\epsilon$$
For $\delta \rightarrow 0$, since $\epsilon$ is arbitrary, we have the result.
