On Wikipedia, a Borel measure is defined as any measure that is defined on the Borel $\sigma$-algebra of a locally compact Hausdorff topological space. Obviously if you have a measure $\mu$ that is defined on a $\sigma$-algebra which strictly contains the Borel $\sigma$-algebra, you can restrict it to the Borel $\sigma$-algebra. But would $\mu$ itself, defined on the bigger $\sigma$-algebra, still be called a Borel measure?
As a more general question: in math, when we say "defined on $X$", does that mean "defined on $X$ and nothing else" or "defined at least on $X$"?
Thank you!