I have an adaptet, cadlag proces $\displaystyle M $ with $\displaystyle M(0)=0$. Then I have that the jumps $\displaystyle \Delta M \geq 0$ and a stoppingtime

$\displaystyle
\tau_{x}=inf \{ t \geq 0 : M(t) - \mu t <x \}
$

But why is it that

$\displaystyle M( \tau_{x} )= \mu \tau_{x} + x$

on $\displaystyle (\tau_{x} < \infty) $?

And not $\displaystyle M( \tau_{x} ) < \mu \tau_{x} + x$

Can anyone explain this?