understanding of this $\int_0^{\infty} P[Y>t]dt = \sum_{n=0}^{\infty} \int_{mn}^{m(n+1)} P[Y>t]dt $

Question

My question is about understanding a step of the answer of the following question.

show that $\limsup_{n\rightarrow\infty} \frac{\Sigma_{k=1}^{n} X_k}{n}<\infty$ a.s.

I want to know how the following step has derived in the answer of the above question

$\int_0^{\infty} P[Y>t]dt = \sum_{n=0}^{\infty} \int_{mn}^{m(n+1)} P[Y>t]dt $

Is there any calculus theorem /definition behind this ?

I am very keen to learn mathematical statistics and the understanding of this step will really helps me to solve lot of similar problems.

I cannot directly comment on that question due to my lower reputation.

Answer 1

Well.

Observe \begin{align} \int^\infty_0 f(t)\ dt = \int_0^m f(t)\ dt+\int_{m}^{2m} f(t)\ dt+ \int^{3m}_{2m} f(t)\ dt + \ldots \end{align}