Does the Dirac measure still exist on a complex domain?

[friend]

does the Dirac measure still exist with complex variance?

The Dirac delta function can be rigorously defined as a measure. See

http://en.wikipedia.org/wiki/Dirac_delta_function#As_a_measure


For the gaussian form of the Dirac delta function we have,

$$\[<br /> {\rm{\delta (x - x}}_0 ) = \mathop {\lim }\limits_{\Delta \to 0} \frac{1}{{(2\pi \Delta )^{1/2} }}e^{ - (x - x_0 )^2 /(2\Delta )} <br /> \]<br />$$

with variance $$\Delta$$.

My question is whether the gaussian from of the Dirac delta is still a measure if the variance is complex. Any insight about this would be appreciated.

 

[fresh_42]

What do you mean by complex variance? We need a measurable space and a $\sigma-$algebra. The space can be a complex vector space. Only the measure has to be a function into $[0,1]\subseteq \mathbb{R}$. A volume cannot be complex.

See https://en.wikipedia.org/wiki/Dirac_measure.