Brenden Nyhan says it can't be measured directly:
The problem is that misperceptions are not necessarily the result of a closed information loop. Someone with a relatively balanced media diet can still end up with false beliefs -- it all depends on how they interpret the news that they receive (i.e., the extent to which they are willing to accept information that is inconsistent with their preferences).
A better approach would be to measure (a) to what extent ideological elites on the left and right are failing to engage with outside sources of information and (b) to what extent their adherents are consuming political news largely or entirely from like-minded elites.
Henry Farrell counters:
Data on divergent patterns of media and information consumption is valuable in figuring out what people think. But people interested in this question aren’t so much worried about the actual patterns of consumption as about its putative consequences for political beliefs. So I think that first cut research to identify whether epistemic closure is a problem should focus on consequences, contra Brendan, looking at the extent to which individuals with different ideologies tend towards closure across a variety of politically salient issues. But it would be nice to see a second wave of research, extending the stuff that Brendan talks about to look at how variation in patterns of media consumption intersected with false political beliefs. And a third body of research could do some experimental work to figure out more precisely the underlying causal mechanisms …