General Definition of Likelihood Function

I am wondering if there is a way of generalizing the likelihood function of some parameters $L(\theta | \mathbf x), \theta \in \Theta$ given some data $\mathbf x$ that has been observed coming from a distribution on $\mathbb R^n$, say $\mu$, so that it is defined for distributions that admit neither a density nor mass function. I guess something rooted in measure theory that preserves the relevant properties of likelihood functions would be desired (e.g. under suitable regularity conditions it would be desirable for asymptotic normality and consistency of maximum likelihood estimators to hold).

My only thought is that we may be able to generalize the notion so long as the distribution of the data admits some density with respect to some measure other than Lebesgue or counting measure.

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Well, this is believable. Do you know off-hand if any good results such as $\sqrt n$-consistency of MLE's hold under reasonable regularity conditions? –  guy Jul 16 '11 at 22:08