User:IssaRice/Lebesgue theory

some questions for now:

• why all the asymmetry in the usual definitions? [1]
• why isn't the lebesgue integral defined as the area under the graph? pugh's book does it this way. why is the definition in terms of simple function or the inf thing that axler does in MIRA preferred by textbooks?
• what would a corresponding "riemann measure" look like for subsets of R^n? is that just the jordan measure?
• why is caratheodory's criterion for measurability defined the way it is? there was a good blog post i saw once that gave a picture but i don't remember if i was fully convinced.
• is the only difference between jordan and lebesgue measure that one has a finite number of boxes and the other has countably many boxes? seems like it [2]
  • in that case, one question i have is, why can't we reach the lebesgue integral simply by taking partitions along the x-axis with countably many points, instead of finitely many points (as in the riemann integral)?
• related to the asymmetry question: why can't we define a set to be lebesgue measureable iff its outer and inner lebesgue measures coincide, just like with jordan measurability? it must be that caratheodory's definition generalizes better. so there must be a theorem like "if a set is bounded, then the outer and inner lebesgue measure coincide if and only if caratheodory's criterion is satisfied". then this theorem justifies using the caratheodory criterion to try to measure unbounded sets.

pugh's book has more connections between riemann and lebesgue