In the context of probability theory, I have come across $\int_{[a,b]} f(x)\ dQ(x)$ for some function $f:\mathbb{R}\to\mathbb{R}$ and some measure $Q(x) = \exp(-\lambda \cdot x)$. Since $\frac{dQ(x)}{dx} = -\lambda \cdot \exp(-\lambda \cdot x)$ we can write $$\int_{[a,b]} f(x)\ dQ(x) = -\int_{[a,b]} f(x)\cdot\lambda\cdot \exp(-\lambda \cdot x)\ dx$$
Questions:
- Theoretical Side: I am not quite sure how $Q(x)$ defines a "measure", e.g. what is the measure of $[a,b]$? Is it $Q(b) - Q(a)$? What's the correct way of labeling $Q(x)$ and $dQ(x)$?
- Practical Side: What if $Q(x)$ is a function (different to the one above), to which I don't know the derivative $\frac{dQ(x)}{dx}$. How can I numerically approximate the integral? Would this work: $$\int_{[a,b]} f(x)\ dQ(x) \quad\approx\quad \sum_{i=0}^{N-1} \frac{f(x_{i+1}) + f(x_{i})}{2} \cdot \Big(Q(x_{i+1}) - Q(x_{i})\Big)$$ for some partition $a=x_0<\ldots < x_N=b$ ?
- Are there better methods to numerically integrate than the method from (2)?