3 ms·

hear hear. lesbesgue measure. stuff of nigtmares for me

graycat 3y ago
Sorry! I LOVE measure theory and find it much better and especially EASIER than traditional Riemann integration!!! In simplest terms, do the partitioning on the Y-axis (the range of the function being integrated) instead of the X-axis (the domain). Right away that makes good sense because we KNOW the range is just the real numbers, and partitioning on the range frees up the domain to be quite general, with a sigma algebra, e.g., where do the sampling as in probability theory. In one step further, get to f'get about the domain of the function being a compact set and the function being continuous and uniformly continuous -- just get to set all that aside. Also get a straight forward theory of the domain being, say, the whole real line without any extra work, without the Riemann approach where have to take a limit as the limits of the integral go to infinity. And, again compared with Riemann, the measure theory limit that defines the integral is simpler and makes more sense. For an extra goodie, get a nice, unusually general proof of Leibniz's rule, that is, differentiation under the integral sign! W. Rudin's Real and Complex Analysis has a really nice, succinct, clean (precisely reasoned) treatment early in that book. Maybe what I have explained here can provide enough of an overview and some intuition to make the Rudin material quite easy to read.
tech_ken 3y ago
Lebesgue measure is a lot of annoying terminology, but the end result are basically integrals that work like how you intuitively "want" them to work. Basically so long as you never have to evaluate one using the actual definition then you're gucci, just pretend like you're a physicist and think about it terms of things like point-masses.
rq1 3y ago
Lebesgue integration is rather simple. You should read the article-correspondence between Lebesgue and Picard. The “entire” construction literally fits on two pages. It’s pure genius.
- mhh__ 3y ago
  > You should read the article-correspondence between Lebesgue and Picard. Where?
  - rq1 3y ago
    https://perso.lpsm.paris/~mazliak/Lebesgue_1901.pdf https://perso.lpsm.paris/~mazliak/Lebesgue_1901.pdf 4 pages*
    - mhh__ 3y ago
      Thanks

Sorry! I LOVE measure theory and find it much better and especially EASIER than traditional Riemann integration!!! In simplest terms, do the partitioning on the Y-axis (the range of the function being integrated) instead of the X-axis (the domain). Right away that makes good sense because we KNOW the range is just the real numbers, and partitioning on the range frees up the domain to be quite general, with a sigma algebra, e.g., where do the sampling as in probability theory. In one step further, get to f'get about the domain of the function being a compact set and the function being continuous and uniformly continuous -- just get to set all that aside. Also get a straight forward theory of the domain being, say, the whole real line without any extra work, without the Riemann approach where have to take a limit as the limits of the integral go to infinity. And, again compared with Riemann, the measure theory limit that defines the integral is simpler and makes more sense. For an extra goodie, get a nice, unusually general proof of Leibniz's rule, that is, differentiation under the integral sign! W. Rudin's Real and Complex Analysis has a really nice, succinct, clean (precisely reasoned) treatment early in that book. Maybe what I have explained here can provide enough of an overview and some intuition to make the Rudin material quite easy to read.

tech_ken 3y ago

Lebesgue measure is a lot of annoying terminology, but the end result are basically integrals that work like how you intuitively "want" them to work. Basically so long as you never have to evaluate one using the actual definition then you're gucci, just pretend like you're a physicist and think about it terms of things like point-masses.

rq1 3y ago

Lebesgue integration is rather simple. You should read the article-correspondence between Lebesgue and Picard. The “entire” construction literally fits on two pages. It’s pure genius.

mhh__ 3y ago
> You should read the article-correspondence between Lebesgue and Picard. Where?
- rq1 3y ago
  https://perso.lpsm.paris/~mazliak/Lebesgue_1901.pdf https://perso.lpsm.paris/~mazliak/Lebesgue_1901.pdf 4 pages*
  - mhh__ 3y ago
    Thanks