Why $\int_{-\infty}^x\sum_{i=1}^{+\infty} p^{i-1}\delta(\alpha-i)d\alpha = \sum_{i=1}^{+\infty}\int_{-\infty}^x p^{i-1}\delta(\alpha-i)d\alpha$
where $\delta$ is the Dirac's delta and $p \in ]0;1[$ is a number ?
Why the infinite series and the integral they can be exchanged ? Which theorems are involved ?
Thanks!
If $p$ is a number, you can push it out of the integral, and then it's a trivial theorem about discrete series.