Proof that $E[\exp(|A|)] \leq \exp(\delta) + \exp(\delta)P(|A|\delta) + \int_\delta^\infty\exp(x)P(|A|x)dx$

Proof that $E[\exp(|A|)] \leq \exp(\delta) + \exp(\delta)P(|A|\delta) +
\int_\delta^\infty\exp(x)P(|A|x)dx$

pLet $A$ be a random variable and $P$ be a probability measure. For some
real $\deltagt;0$, is there a simple proof that $E[\exp(|A|)] \leq
\exp(\delta) + \exp(\delta)P(|A|gt;\delta) +
\int_\delta^\infty\exp(x)P(|A|gt;x)dx$?/p pThis statement was written as
obvious by a href=http://link.springer.com/article/10.1007/BF01213388
rel=nofollowCapitaine/a, page 196 and a
href=http://rads.stackoverflow.com/amzn/click/0444861726 rel=nofollowIkeda
and Watanabe/a page 450, but I've failed to get any progress in
understanding it. Capitaine says it follows from Fubini's theorem. I think
it is not necessary to assume that $A$ admits a density./p