Wikipedia should have said that $L(\theta)$ is not a conditional probability of $\theta$ being in some specified set, nor a probability density of $\theta$. Indeed, if there are infinitely many values of $\theta$ in the parameter space, you can have $$ \sum_\theta L(\theta) = \infty, $$ for example by having $L(\theta)=1$ regardless of the value of $\theta$, and if there is some standard measure $d\theta$ on the parameter space, then in the same way one can have $$ \int_\Theta L(\theta)\,d\theta =\infty. $$ An essential point that the article should emphasize is that $L$ is the function $$ \theta \mapsto P(x\mid\theta) \text{ and NOT } x\mapsto P(x\mid\theta). $$

Wikipedia should have said that $L(\theta)$ is not a conditional probability of $\theta$ being in some specified set, nor a probability density of $\theta$. Indeed, if there are infinitely many values of $\theta$ in the parameter space, you can have $$ \sum_\theta L(\theta) = \infty, $$ for example by having $L(\theta)=1$ regardless of the value of $\theta$, and if there is some standard measure $d\theta$ on the parameter space $\Theta$, then in the same way one can have $$ \int_\Theta L(\theta)\,d\theta =\infty. $$ An essential point that the article should emphasize is that $L$ is the function $$ \theta \mapsto P(x\mid\theta) \text{ and NOT } x\mapsto P(x\mid\theta). $$