Let $\mu : \mathbb{B(\mathbb{R})} \to \mathbb{R}$ such as $$\forall A \in \mathbb{R},\, \mu(A) = \frac{1}{4}\int_A e^{-|x|} \, \mathrm{d}x \, + \frac{1}{2}\mathbb{1}_A(0)$$
Let $f : \mathbb{R} \to \mathbb{R}$ a positive Borel function.
I followed a course on Measure Theory but we never integrated with something else than Lebesgue measure. I'm now doing a probability exercice in which I have to give a formula in order to calculate $\int_\mathbb{R} f(x)\, \mathrm{d}\mu(x)$. I have the answer right in front of my eyes but still can't manage to understand how we integrate with respect to $\mu$
EDIT : Is it correct ? : $$\int_{\mathbb{R}} f(x)\, \mathrm{d}\mu(x) = \require{cancel}\cancel{\int_{\mathbb{R}} f(x)\mu(\{x\})\, \mathrm{d}x} = \frac{1}{4}\int_{\mathbb{R}} f(x)e^{-|x|}\,dx +\frac{1}{2}f(0)$$