What is meant by finite action signal
It's something like "finite energy signals"

$$ \int_{-\infty}^{\infty}|x(t)|^2 dt&lt;\infty $$

$$ \sum_{n=-\infty}^{\infty}|x[n]|^2 &lt;\infty $$

except that the "finiteness" is a tougher constraint with

$$ \int_{-\infty}^{\infty}|x(t)| dt&lt;\infty $$

$$ \sum_{n=-\infty}^{\infty}|x[n]| &lt;\infty $$

for example, the signal

$$ x(t) = \frac{1}{\sqrt{1+t^2}} $$

is finite energy, but is not finite action.