Question: I know that many Statistics PhD students come from undergraduate / master's level math / applied math backgrounds. So, many will have taken a full-year of introductory real analysis and most likely also complex analysis (and lots of linear algebra and linear algebraic courses, such as scientific computing) -- but perhaps not measure-theoretic analysis.

Would you say that it is imperative to try and take a measure theory course to strengthen one's application to PhD programs in Statistics?

A lot of my classmates are fixated on the idea that measure theory is a necessity for Phd admission, while one professor that I spoke with internally thinks that this is a myth. But, he is a mathematician, so I'd rather ask the question here, just to gather more information.

Thanks,

Answer: 
Measure theory is essential for higher level probability.

However, I believe most graduate schools will have a graduate course on measure theory. Thus it is not absolutely necessary to have learnt measure theory before entering graduate school.
Most statistics doctoral programs don't require their applicants to know measure theory, but then again, some very good doctoral statistics programs don't even require measure theory at all. (Case in point is Virginia Tech, which at least back in the early 2000's when I was there, had a great applied statistics department and measure theory was not required for the PhD program). 

Do you need measure theory do statistics? Well, if you plan to work in academics as a statistician or want to do cutting edge research in financial analysis, you probably need it but I would say for a large portion of applied statisticians working outside of academics, the answer is no. IMO, it is not that important for the vast majority of what applied statisticians do (experimental design, data visualization, and estimation). Most of my friends who have high level positions in companies as statistical consultants report they never use it (which has been true for me also).