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Consider a task of estimating a parameter of a censored exponential distribution using maximum likelihood estimation. The typical approach to this question is presented in this question.

The linked answer asserts that $$P_\lambda(X = x_i) = \left \{ \begin{array}{rl} \lambda e^{-\lambda x_i} & \text{, if } x_i \leq t_i\\ e^{-\lambda t_i} & \text{, if }x_i > t_i\\ \end{array}\right.$$ is the likelihood function of the censored distribution. While it seems intuitive, the problem is that the distribution of $X$ is neither continuous nor discrete, so we can't apply the standard notions of likelihood.

Does the concept of likelihood generalize to such distributions? If yes, what's the general definition, and how does it apply to the distribution of $X$.

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