I'm trying to understand this problem but it does not go very well. I don't understand how I should calculate this integral, when I'm given the density function. Read the problem below:
"Let $\lambda$ be a Borel measure on the interval $[0;\infty)$ given by the density function $\exp(-x)$ with respect to the 1-dim. Lebesgue measure. Evaluate the following integral $$\begin{equation} \int_{[0;\infty)} \sin(x)d\lambda(x) \end{equation}$$ "
Should I consider the problem as follows $$\begin{equation} \int_{[0;\infty)} e^{-x}\sin(x)d\lambda(x) \end{equation}$$ Because if I solve the integral by Riemann, then I get the pretty result as:$$\frac{1}{2}$$
Am I on the right track? Thanks in advance.