I'm applying to math PhD programs in the US (I'm also in the US). In my statement of purpose, I write that I want to specialize in probability theory. However, this semester I took a measure-theoretic course in probability theory, and I did not do well on the final exam. This resulted in a 3.5 final grade (at my school, we have 4.0 then 3.5, 3.0, etc).
I worry that getting a 3.5 in the subject that I said I wanted to specialize in would have a negative impact on my application, especially for the top schools. Would it be wise to revise my statement of purpose and replace probability with something else I'm interested in? I'm broadly interested in analysis.
Also, I submitted the transcript that only has the courses I took before the Fall semester, but I they might ask about my Fall grades at some point during the review process (or during an interview).
Thank you!
Edit: added the country