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bilateral Laplace transform of a non-finite borel measure $\mu$, is it possible?

$$\int_{-\infty}^{+\infty} e^{-xt}\, \mathrm d\mu(t),\,\, x\in \mathbb{R}$$

If make sense to define a bilateral laplace transform of a non-finite measure, what changes in the region of convergence? That $x=0$ is not necessarly included in the region of convergence?

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  • $\begingroup$ Take a look at the Wiki article. $\endgroup$
    – user64494
    Commented Jun 17, 2013 at 11:08
  • $\begingroup$ "In terms of the Fourier transform, we may also obtain the two-sided Laplace transform, as The Fourier transform is normally defined so that it exists for real values; the above definition defines the image in a strip which may not include the real axis." So why if fourier transform is always defined for real values, the strip of convergence of the laplace transform may not include the real axis? $\endgroup$
    – alemou
    Commented Jun 17, 2013 at 11:20

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