"Dominance" condition for consistency of MLE

On the Wikipedia page for maximum likelihood estimation, there are a set of four conditions listed for the consistency of the maximum likelihood estimator. I had a question about the last one, called "dominance" in that article:

There exists $D(x)$ integrable with respect to the distribution $f(x | theta_0)$ such that $$ {big |}ln f(xmid theta ){big |}<D(x)quad {text{ for all }}theta in Theta .$$

Does this mean that if the likelihood function goes to zero ($f(x|theta) = 0$) anywhere on $Theta$, this condition isn’t valid since $|ln (x|theta)| to infty$ at such points? Are there weaker conditions that would hold for a likelihood function which vanishes at some points in parameter space, but still lead to the MLE being consistent?

If a toy example would help, consider the likelihood function
$$
f(x | theta) = begin{cases} frac{1}{2 theta} cos(x/theta) & |x|leqpitheta/2 \
0 & |x|>pi theta/2 end{cases}
$$
with $theta in Theta = [a,b]$ for some $a, b > 0$.
Would applying the MLE method to a sample drawn from this distribution lead, in the limit of a large number of data points, to an accurate estimate of $theta$?