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.
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.