Question

can anyone please show me that how to apply Lebesgue integral by showing some example like we do Riemann integral in calculus class?

Answer 1

Hope someone will answer... I would like to see it too... (sorry for SPAM)

Answer 2

can someone show in the similar way we did normal integration such as http://media.wiley.com/Lux/72/39572.nce016.gif

Answer 3

@KingGeorge any clue bro?

Answer 4

Sorry, can't help you on this one :/

@amistre64 @satellite73

Answer 5

the truth is that no one want to do a reimann integral either, because even simple examples are hard.
also, if a reimann integral exists, it is the same as a lebesgue integral.
the only difference is that integrating over a measure instead of intervals allows one to make the space of integrable functions "complete" as in if you have a sequence of integrable functions that converges, the limit function also must be integrable.
not so with reimann integrals, classic example being the characteristic function of the rationals

Answer 6

if you think about how you compute a reimann integral, there is no magic. you either cook it up so that you end up with some nice summation formula you can use, or else you appeal to the fundamental theorem of calculus and compute via anti - derivatives. even a simple example like \[\int_0^\pi \sin(x)dx\] you would not compute as a limit of a reimann sum, you would compute via fundamental theorem. otherwise it would be a numeric, computer calculation

Answer 7

how to simply solve x or x^2 with lebesgue

Answer 8

you would compute the same way you compute a riemann integral because measure on the real line is the same as the interval

Answer 9

in other words you would divide the length of the path in to subintervals just like with a riemann integral and then compute the limit of a sum

Answer 10

really? but in an analysis book it says that it integrates the range instead of domain like Riemann does

Answer 11

range means i believe range of the measure, but the measure on the real line is just the length of the intervals, so it is the same

Answer 12

& it divides the y-axis horizontally instead of x-axis vertically

Answer 13

really? man i hope i am not completely wrong, but this is what i remember.
try googling an example and see what you get

Answer 14

from wikipedia: http://upload.wikimedia.org/wikipedia/commons/1/1b/RandLintegrals.png

Blue Riemann, Red Lebesque

Answer 15

"Lebesgue integration is to partition the range of the function f rather than its domain" Kolmogorog's intro. real analysis.
http://en.wikipedia.org/wiki/File:RandLintegrals.png
http://en.wikipedia.org/wiki/Lebesgue_integration#Introduction please read the letter Lebesgue wrote to his student Paul

Answer 16

@satellite73 unfortunately I can't find a single damn' example of it, all the damn' examples show abstract work (the one we find in any analysis book) :( it will be really great if someone can share a link with me.

Answer 17

try here
http://rutherglen.science.mq.edu.au/wchen/lnilifolder/lnili.html

Answer 18

not even maple has any solution for it

Answer 19

where should I go on this link?

Answer 20

again the problem is that as i said, how would you compute a riemann integral? you would not be able to unless you have a simple example
and further it is for sure true that if a function is riemann integrable is it lebesque integrable and the integrals are the same, so only examples you ever see of actual lebesgue integrals in action are the odd examples of functions that are lebesque but not riemann integrable

Answer 21

gotta run eat but if i can find some example in action i will send it

Answer 22

enjoy eating
I will really appreciate if you can post any link

Answer 23

i looked through the line i sent. it actually 4 examples. i am not sure they are that enlightening, but they may be. example 7.4.4 would be good to look at and example 7.4.6 is exactly the question you asked above, for \(x\) as well as for \(x^2\)