Cool post! I like this both because finding examples is the best way to work out what a definition constrains and captures, and also because I’m studying a bit of measure theory too.
So far as I can tell, the Jordan measure and Borel measure are very similar to the Lebesgue measure, and as such I struggle to find real-world examples of them that explain how they are different from the Lebesgue. If someone puts intuitive examples of these in the comments, I’ll add them to this post.
Checking quickly on Wikipedia, it seems that the Borel measure is literally the same as the Lebesgue measure, but defined on a smaller σ-algebra (the Borel σ-algebra instead of the Lebesgue σ-algebra, see here). As for the Jordan “measure”, it is apparently a precursor of the Lebesgue measure, and is not a true measure as it’s not defined on a σ-algebra (see here). The difference is that the Jordan “measure” is only defined on sets with boundary of Lebesgue/Jordan measure 0, which is not the case for the Lebesgue measure apparently.
All that to say, not having examples of Borel and Jordan measure is not an issue, as these would apparently be limited Lebesgue measures anyway.
That helps—I wasn’t sure whether there might maybe be some small special intuitive difference in Borel or Jordan that could correspond to a different real world example, but now I think that’s definitely a No.
Cool post! I like this both because finding examples is the best way to work out what a definition constrains and captures, and also because I’m studying a bit of measure theory too.
Checking quickly on Wikipedia, it seems that the Borel measure is literally the same as the Lebesgue measure, but defined on a smaller σ-algebra (the Borel σ-algebra instead of the Lebesgue σ-algebra, see here). As for the Jordan “measure”, it is apparently a precursor of the Lebesgue measure, and is not a true measure as it’s not defined on a σ-algebra (see here). The difference is that the Jordan “measure” is only defined on sets with boundary of Lebesgue/Jordan measure 0, which is not the case for the Lebesgue measure apparently.
All that to say, not having examples of Borel and Jordan measure is not an issue, as these would apparently be limited Lebesgue measures anyway.
Thanks!
That helps—I wasn’t sure whether there might maybe be some small special intuitive difference in Borel or Jordan that could correspond to a different real world example, but now I think that’s definitely a No.