Decomposition of the probability of the sum

Question

I cannot understand how is gotten the following decomposition.

Supposing that $X_1,...,X_n$ random variables i.i.d with heavy tailed distribution $S_n=\sum_{i=1}^nX_i$

In the article that I m reading is stated $$P(S_n>u )=P(\max_{i\leq n} X_i>u)+P(Sn>u,\max_{i\leq n}X_i\leq u)$$ Why $Sn>u$ and $\max_{i\leq n}X_i\leq u$ are independent? the article is the next one, in the page 2.

Answer 1

They are breaking down $\{S_n>u\}$ into two disjoint events: either $\max_{i\leq n}X_i>u$ or it isn't. Presumably $X_i\geq 0$, so that clearly if the maximum exceeds $u$ then so does $S_n$. Alternatively all $X_i$ might be less than $u$, in which case you need $S_n>u$.

Note that $X_i\geq 0$ is required, otherwise this is false.