How to rigorously define the likelihood? The likelihood could be defined by several ways, for instance :


*

*the function $L$ from $\Theta\times{\cal X}$ which maps $(\theta,x)$ to $L(\theta \mid x)$ i.e. $L:\Theta\times{\cal X} \rightarrow \mathbb{R} $.

*the random function $L(\cdot \mid X)$

*we could also consider that the likelihood is only the "observed" likelihood $L(\cdot \mid x^{\text{obs}})$

*in practice the likelihood brings information on $\theta$ only up to a multiplicative constant, hence we could consider the likelihood as an equivalence class of functions rather than a function
Another question occurs when considering change of parametrization: if $\phi=\theta^2$ is the new parameterization we commonly denote by $L(\phi \mid x)$ the likelihood on $\phi$ and this is not the evaluation of the previous function $L(\cdot \mid x)$ at $\theta^2$ but at $\sqrt{\phi}$. This is an abusive but useful notation which could cause difficulties to beginners if it is not emphasized.
What is your favorite rigorous definition of the likelihood ? 
In addition how do you call $L(\theta \mid x)$ ? I usually say something like "the likelihood on $\theta$ when $x$ is observed".
EDIT: In view of some comments below, I realize I should have precised the context. I consider a statistical model given by a parametric family $\{f(\cdot \mid \theta), \theta \in \Theta\}$ of densities with respect to some dominating measure, with each $f(\cdot \mid \theta)$ defined on the observations space ${\cal X}$. Hence we define $L(\theta \mid x)=f(x \mid \theta)$ and the question is "what is $L$ ?" (the question is not about a  general definition of the likelihood)
 A: I think I would call it something different.  Likelihood is the probability density for the observed x given the value of the parameter $θ$ expressed as a function of $θ$ for the given $x$.  I don't share the view about the proportionality constant.  I think that only comes into play because maximizing any monotonic function of the likelihood gives the same solution for $θ$.  So you can maximize $cL(θ∣x)$ for $c>0$ or other monotonic functions such as $\log(L(θ∣x))$ which is commonly done.
A: Here's an attempt at a rigorous mathematical definition:
Let $X: \Omega \to \mathbb R^n$ be a random vector which admits a density $f(x | \theta_0)$ with respect to some measure $\nu$ on $\mathbb R^n$, where for $\theta \in \Theta$, $\{f(x|\theta): \theta \in \Theta\}$ is a family of densities on $\mathbb R^n$ with respect to $\nu$. Then, for any $x \in \mathbb R^n$ we define the likelihood function $L(\theta | x)$ to be $f(x | \theta)$; for clarity, for each $x$ we have $L_x : \Theta \to \mathbb R$. One can think of $x$ to be a particular potential $x_{obs}$ and $\theta_0$ to be the "true" value of $\theta$. 
A couple of observations about this definition:


*

*The definition is robust enough to handle discrete, continuous, and other sorts of families of distributions for $X$.

*We are defining the likelihood at the level of density functions instead of at the level of probability distributions/measures. The reason for this is that densities are not unique, and it turns out that this isn't a situation where one can pass to equivalence classes of densities and still be safe: different choices of densities lead to different MLE's in the continuous case. However, in most cases there is a natural choice of family of densities that are desirable theoretically.

*I like this definition because it incorporates the random variables we are working with into it and, by design since we have to assign them a distribution, we have also rigorously built in the notion of the "true but unknown" value of $\theta$, here denoted $\theta_0$. For me, as a student, the challenge of being rigorous about likelihood was always how to reconcile the real world concepts of a "true" $\theta$ and "observed" $x_{obs}$ with the mathematics; this was often not helped by instructors claiming that these concepts weren't formal but then turning around and using them formally when proving things! So we deal with them formally in this definition.

*EDIT: Of course, we are free to consider the usual random elements $L(\theta | X)$, $S(\theta | X)$ and $\mathcal I(\theta | X)$ and under this definition with no real problems with rigor as long as you are careful (or even if you aren't if that level of rigor is not important to you).

A: Your third item is the one I have seen the most often used as rigorous definition. 
The others are interesting too (+1). In particular the first is appealing, with the difficulty that the sample size not being (yet) defined, it is harder to define the "from" set.
To me, the fundamental intuition of the likelihood is that it is a function of the model + its parameters, not a function of the random variables (also an important point for teaching purposes). So I would stick to the third definition.
The source of the abuse of notation is that the "from" set of the likelihood is implicit, which is usually not the case for well defined functions. Here, the most rigorous approach is to realize that after the transformation, the likelihood relates to another model. It is equivalent to the first, but still another model. So the likelihood notation should show which model it refers to (by subscript or other). I never do it of course, but for teaching, I might.
Finally, to be consistent with my previous answers, I say the "likelihood of $\theta$" in your last formula.
