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.
- is there an analogue of the caratheodory theorem for jordan measure, to allow us to extend jordan measurability to unbounded sets? or is this not an interesting question to ask since if we try to measure such sets, the answer will always be infinity even for "thin" sets like Q?
i've always found the coin-counting analogy between riemann and lebesgue measure confusing. why should one method be better than the other, when they both produce the same answer? if i was actually trying to quickly estimate how much i had in coins, i would just gather up all the highest denominations and count those, then add a "fuzz factor" to account for some error. i wouldn't even bother counting the pennies unless it seemed like there was a huge number of them.
the riemann/jordan vs lebesgue difference has been described as finite vs countable, x-axis partitioning vs y-axis partitioning, topological boundary measure = 0 vs measure-theoretic boundary measure = 0, and maybe one other thing i am forgetting. but what is the essence of the difference here? why do all these separate distinctions turn out to be "the same thing"?
- why should partitioning the y-axis lead to being able to integrate more functions compared to partitioning the x-axis? the finite vs countable distinction makes sense, but the y vs x thing makes no sense to me.
Apostol's analysis lists two non-equivalent definitions of riemann integrability i think. which one does the jordan-undergraph riemann integral pick out and why?
since there is a notion of riemann sums, is there also an analogous notion of "lebesgue sums"? Yes; see pugh's book.
pugh expresses riemann integrability in terms of the boundary of the function having zero lebesgue measure. why do we have to bring in lebesgue measure here? can't it be jordan measure?
why can't we extend jordan measurability to unbounded sets by doing something analogous to improper riemann integration? like, we define a "measure" for some finite portion of the set parametrized by some bound, then take the limit as the bound goes to infinity.
more general way of generating questions: there's some stuff we talk about a lot in "riemann/jordan land" (e.g. upper and lower sums and defining integrability when the two are equal), and some stuff we talk about a lot in "lebesgue land" (e.g. caratheodory criterion). for each thing we talk about in one of the lands, what is the analogue of it in the other land?

pugh's book has more connections between riemann and lebesgue

also, i am scared to ask, but how does all of this apply to the gauge integral?

i think it's pretty bad that there seems to be no book that answers all of these questions, period, let alone in an easily understandable manner. you can tell these questions are not even asked in the textbooks because professional mathematicians are asking them on mathoverflow... e.g. [3] [4]