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, piecewise constant function vs simple function 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?
• if the lebesgue integral was just the riemann integral but dividing along the y-axis instead of the x-axis, then we should just be able to get the lebesgue integral by integrating f^-1 or something?
• the horizontal stacking picture people like to draw for the lebesgue integral seems deceptive? like, if you actually look at how a simple function is integrated, you kind of draw the horizontal "window" but only to find where the corresponding points are on the x-axis? and then you multiply the y value with the width along the x-axis, so the actual area for that bit is still a vertical rectangle. thank god, someone else noticed this: "I think you haven't understood the Lebesgue integral at all. The animation with horizontal rectangles is quite flawed. That's not how it works!! Rectangles are vertical, what happens is that we split the range of the image into different sections. In each of them, we take an arbitrary point, and multiply it by the inverse function of the section. That may be one or more VERTICAL rectangles."
• my comment: "If the only difference between the Riemann and Lebesgue integral was dividing up along the x-axis vs y-axis, then the Jordan measure (which uses boxes and doesn't care about the x or y axis) should be able to find the area under the graph of any Lebesgue-integrable function, right? The fact that this is not possible I think means there is some other deeper difference between the two integrals."
• trying to measure the set of irrational numbers in [0,1]: i think this should have length 1, but even though it's a bounded set, the sup of the inner measure seems to be 0? actually it does seem to be possible to construct a positive measure subset that only contains irrationals [3], which is pretty unintuitive. hmm, but this is still different from inner measure i think, because we aren't using a countable number of boxes inside the irrationals.. instead we're starting by surrounding the rationals and then substracting out those sets.
• ok here's another reason i think the y-axis thing is bullshit: we could define a "jordan integral" by taking the definition of the lebesgue integral via simple functions but replacing the measure with the jordan measure. i think the resulting integral would be equivalent to the riemann integral.
• if all lebesgue did was to replace "finite" with "countable" in the definition of jordan measure, why was lebesgue's theory considered so important/revolutionary? was it the fact that lebesgue also did all the legwork to prove that his measure/integral had all these nice properties?
• what if we used simple functions to define the riemann integral, or piecewise constant functions to define the lebesgue integral?
• i think a lot of definitions of the lebesgue integral "cheat" by using the lebesgue measure (it puts all the difficulty of measurement into the lebesgue measure part). the riemann integral doesn't use the jordan measure in most definitions; you just add up the rectangles yourself. can we just get the lebesgue integral by using a countable number of rectangles? or is there more to lebesgue than that? can we make use of measurability without using the measure itself, to define the lebesgue integral?
• the horizontal slab idea is coming from this definition. the weird thing though is that almost no one defines the lebesgue integral that way??? so like, the image you present has nothing to do with the technical definition you give. also, the horizontal slab thing is just using the lebesgue measure! all the work is being done by the lebesgue measure! if you substituted the jordan measure instead then i think you would just get back the riemann integral again.

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. [4] [5]