Relation between Laplace transform and measures.

Question

The Laplace transform of a measure $\mu$ on the real line is defined by $$f_{\mu}(s)= \int_{\mathbb{R}}e^{-st}d\mu(t), \hspace{1cm} \forall s \geqslant 0.$$ My question is ----

1)Does the Laplace transform of a measure (finite or infinite) always exists?

2)If not, can it be said that the Laplace transform of a probability measure always exists?

If the support of the measure is changed from the real line to the non-negative part of the real line, what happens to question (1) and (2)?

Answer 1

Your definition of the Laplace transform will not converge in general, because if $\mu$ has support on the whole real line then the exponential term can blow up, depending on the exact form of $\mu$.

The common definition of the Laplace transform is for measures supported on the non-negative real line. (Alternatively you may want to look into the definition of the two-sided Laplace transform, where $t$ in the exponential is replaced by $|t|$.)

Now, consider a $\mu$ that is supported on the non-negative real line, and that is w.l.o.g. positive. Since $e^{-st} \le 1$ for $\Re[s]>0$, it is easy to see that

$|f_\mu(s)| \le f_\mu(0) = \int\limits_0^\infty d\mu(t) = \| \mu \|$.

Hence, if the measure is finite, then its Laplace transform exists. In conclusion, the Laplace transform of a probability measure always exists. (It is in fact (related to) the measure's moment generating function.)

For infinite measures, one needs some regularity conditions on $\mu$ that ensure that the integral does not blow up. One of these would be for example, that its distribution function $F(t) := \mu([0,t))$ is exponentially bounded. That means:

$ \exists K,C > 0: \;\; |F(t)| \le C e^{Kt}$

Then one can show that the Laplace transform exists for $\Re[s]>K$.

Answer 2

If $\mu$ has density $\frac 1 {\pi (1+x^{2})}$ the the Laplace transform exists only for $s=0$.

Answer 3

When the integral (with respect to the parameter $t$) is over the entire real line it is customary to talk about the Fourier transform. A related notion is that of the characteristic function and not of the moment generating function.

Elaborating on Kavi Rama Murphy response, the Laplace transform of a measure $\mu$ with Cauchy density does not exist on the non-negative real half line but only trivially at $t=0$. The non-existence can perhaps be intuited -if not deduced- from the fact that the first moment of $\mu$ is infinite. I would imagine that for infinite measure the Laplace transform does not exist unless there are certain regularity conditions.