Let $I \subseteq \mathbb R$ be a limited interval, and define the measure of $I$ to be $$m(I) := 6\cdot\mathrm{length}(I\cap(3,5)) + 10\cdot\mathrm{count}(I\cap\{3,4,5\})$$ Where $\mathrm{count}$ is the standard counting measure. Evaluate the integral $$\int_{[1,4)} \frac{x}{x^2+1}dm(x)$$
I have managed to evaluate the measure of the domain of integration (it's $16$), but it defies me how I should perform the actual integration with respect to the given measure. After all, as far as I know, I can apply the FTC only with respect to the standard measure on $\mathbb R$—that is, $\mathrm{length}(\cdot)$ itself.
Any clarification would be welcome.