Associativity of the Lebesgue-Stieltjes integral

Question

Let

• $T>0$
• $I:=(0,T]$
• $A$ be right-continuous and of bounded variation
• $f:I\to\mathbb R$ be Borel measurable with $$\int|f|\:{\rm d}|A|<\infty\tag1$$ and $$g(t):=\int_{(0,\:t]}f\:{\rm d}A\;\;\;\text{for }t\in\overline I$$

If $h:I\to\mathbb R$ is bounded and Borel measurable, then $$\int|h|\:{\rm d}|g|<\infty\tag2$$ and $$\int|hf|\:{\rm d}|A|<\infty\tag3\;.$$ Moreover, $$\int_{(0,\:t]}h\:{\rm d}g=\int_{(0,\:t]}hf\:{\rm d}A\tag4\;\;\;\text{for all }t\in\overline I\;.$$ Now, let $(P,N)$ be a Hahn decomposition of $I$ with respect to ${\rm d}A$. Moreover, let $$P':=\left\{f\ge0\right\}\cap P\cup\left\{f<0\right\}\cap N$$ and $$N':=\left\{f\ge0\right\}\cap N\cup\left\{f<0\right\}\cap P\;.$$

How can we show that $(P',N')$ is a Hahn decomposition of $I$ with respect to ${\rm d}g$?

Note that by $(4)$ $${\rm d}g(E\cap P')=\int1_{\left\{\:E\:\cap\:P'\:\right\}}\:{\rm d}g=\int1_{\left\{\:E\:\cap\:P'\:\right\}}f\:{\rm d}A\tag5\;\;\;\text{for all }E\in\mathcal B(I)\;.$$

Answer 1

Let $E\subset P'$. Then $E = E_1\cup E_2$, where $E_1\subset\{f\ge 0\}\cap P$ and $E_2\subset\{f < 0\}\cap N$. We have $$ dg(E) = dg(E_1)+dg(E_2) = \int_{E_1}f\,dA + \int_{E_2}f\,dA\ge 0, $$ because $f\ge 0$ and $dA\ge 0$ in the first integral, whereas $f < 0$ and $dA\le 0$ in the second integral. Similarly, you show that $dg(E)\le 0$ for $E\subset N'$.