It would be quite difficult to explain the difference without going into the depth of measure theory, but it's quite restrictive for a function to be R-integrable. For example, it should be bounded and the Lebesgue-measure of the set of its discontinuity points should be zero. The pontwise limit of R-intergrable functions is not necessarily R-integrable, and even if it is, its integral is not necessarily equals to the limit of the integrals. So Riemann-integrability and pointwise limit operations are not really interchangeable for R-integrable functions, which is quite a drawback. It is inherently linked to Jordan-measure, which is also quite limited, since very few sets are Jordan-measureable and behaves not so well with countably infinite set operations.
In Lebesgue-measure and Lebesgue-integrability the functions and L-measureable sets have been created in such a way, that limit-operations and countably inifinite set-operations should not leave the set of L-integrable functions or L-measureable sets. To say it differently, the Jordan-measureable sets form a ring, the Lebesgue-measureable sets form a sigma-ring (or sigma-algebra).
A simple example is the Dirichlet-function (the characteristical function of the rational numbers), which is discontinuous everywhere, hence it is not R-integrable on any bounded interval, whilst it's Lebesgue-integrable and its integral is zero (since it is almost everywhere zero). It is because the intersection of the rational numbers with any interval is not Jordan-measureable, but it's Lebesgue-measureable with measure zero. The Riemannian integral is defined in terms of simple Jordan-measureable sets (ie. subintervals of some bounded interval, then we take the limit), whilst for Lebesgue-integral it is defined via Lebesgue measureable sets (one possible approach for nonnegative functions is to approximate the measure of the subgraph of f, or taking the limit of thre integral of step functions approximating f in a monotone increasing way defined on measureable sets).
I suggest that you can get a interesting twist on both Tamás's and Luis's answers if you read the first page of the preface of the following book:
"The Integrals of Lebesgue, Denjoy, Perron, and Henstock"
by Russell A. Gordon.
It is available on Google Books. I ignore whether you can access Google Books from Iran though, but I would be happy to take screen captures and email them to you.
Perhaps the simplest illustration of the differences between the integrals of Riemann and Lebesgue is the following. Imagine that you have a lot of coins of different denominations and you need to count how much money you have. Riemann integral answers this question as follows. He consistently adds dignity of another coin to the amount already recorded. Lebesgue integral first splits the set of all coins on the sets of coins of the same denomination. Then calculates the cost of each of the resulting subsets. That is quite simply. And then finds the sum of the resulting values.
For the usual lebesgue measure on R, any Riemann integrable function is Lebesgue integrable, but a Lebesgue integrable function is Riemann integrable if and only if the set of point where it is not continuous is of measure 0.
Lebesgue Theorem allows to recover a kind of weak continuity almost everywhere of the Lebesgue measurable function since it asserts that for almost any x the limit of $1/2r\int_{x-r}^{x+r}|f(x)-f(y)|dy$ is equal to 0 when r tends to 0.
For a while let us see how we find area of a square. It is the square of the side. The area of a rectangle whose sides are positive integers can be found by splitting the rectangle in to several smaller squares. Of course, the inherent axioms are the following: area of the boundary is zero, area of disjoint squares add up and area is translation invariant. Area of rectangle with rational sides can be suitably dealt with. When the sides are any positive real numbers we enter a limiting process through rational numbers. Thus the area of a rectangle is equal to ab where a is the length and b is the breadth of the rectangle. Area of any quadrilateral or triangle, for that matter any polygon can be found from these premises.
But when we attempt to find area of a bounded region whose boundary no more consists of straight line segments, but consists of curves there is no straight forward answer. Now, for simplicity, imagine that we are trying to find area of a region in the X-Y plane bounded by the lines x = a, x = b, the line y = 0 and the curve y = f( x ). We may call this, if we feel like, a curvilinear rectangle. Because we do not have any ready made tool to find the area of this curvilinear rectangle we try to approximate this area by successively adding area of smaller adjecent rectangles. To be more precise, we first partition the interval [a,b], which is the breadth of our curvilinear rectangle, to smaller successive pieces namely a = a_0 < a_1
There is an intuitive answer to this question that can be found in some of the textbooks on Lebesgue integration. Let f be a function defined on a compact interval [a,b] and suppose that it is bounded. Riemann integrabillity refers to taking partitions x_1
I think one of the most overlooked and simple differences between the Riemann and Lebesgue integral is the fact that the preimage set function is better behaved than the image set function. The preimage commutes with unions and intersections whereas the image set function does not commute with intersections. That is f[A intersect B] is a subset of f[A] intersect f[B] but may not be equal. This is what is hiding behind the scenes that makes Lebesgue measure better.
Let f(x)=1 for rational x and f(x)=-1 otherwise on [0,1]. Then |f|=1 is Riemann integrable on [0,1], but f is not. For Lebesgue integral: f is integrable iff |f| is integrable.
Lebesgue's approach to integration was summarized in a letter to Paul Montel. He writes:
" I have to pay a certain sum, which I have collected in my pocket. I take the bills and coins out of my pocket and give them to the creditor in the order I find them until I have reached the total sum. This is the Riemann integral. But I can proceed differently. After I have taken all the money out of my pocket I order the bills and coins according to identical values and then I pay the several heaps one after the other to the creditor. This is my integral."
—Source: (Siegmund-Schultze 2008)
The slightly humorous view from the applied math/engineering side comes from Richard Hamming "if whether an airplane would fly or not depended on whether some function that arose in its design was Lebesgue but not Riemann integrable, then I would not fly in it." -- quoted from R Hamming "Mathematics on a Distant Planet" (1998) Am. Math. Monthly 105(7), 640-650.
I think that there is an increasing difficulty/generality in the various kinds of integrals:
-.Cauchy´s integral: intregrand continuous , finite integration interval
-Riemann´s integral: integrand can be not continuous on an interval of zero measure. Furthermore, we can integrate from minus infinity to a finite integral limit , from minus to plus infinity or from a finite inferior limit to plus infinity. Those are the so-called improper Riemann´s integrals and are defined as limits. Riemann- Stieljes´s integral is a generalization ( the differential dx can be replaced with dalpha (x)- a differential defined with a function alpha , Riemann´s integral includes Cauchy´s integral.
- Lebesgue´s integral : The integrand can have discontinuities on intervals of nonzero measure. Lebesge´s integral includes Riemann´s inrtegral.
I have a related question : are they included here infinite - type discontinuities ; like finite escapes of the integrands ( the integrand has an asymptote at some finite point or at the limit of an infinite integration interval) or impulses( defined through Dirac´s distributions) or would we better to include them in a generalized Lebesgue´s integral-type or to use another name for those integrals?.
Using the Lebesgue integral, it is possible to integrate some functions that are unbounded in every interval. For example, if f(x)=1/sqrt(x) and {r_n} is an enumeration of the rational numbers, then g(x)=\sum f(x-r_n )/2^n is unbounded in every interval but has a finite integral.
Distributions are a different thing, although the Dirac distribution can be regarded as a measure and the general Lebesgue theory applies to it. The derivative of the Dirac distribution is not a measure, though.
A little bit more general concept, namely the Henstock-Kurzweil (gauge) integral can be useful to distinguish the two integrals. Please check in some books (Henstock, Gordon, Schwabik). Many examples included.
The basic difference between the Riemann integral and Lebesgue integral is following. Riemann integral works for the step function, on the other hand Lebesgue integral works for the simple function.
Also you can compare with the following. The main difference between the Lebesgue and Riemann integrals is that the Lebesgue method takes into account the values of the function, subdividing its range instead of just subdividing the interval on which the function is defined. This fact makes a difference when the function has big oscillations or discontinuities. However, the Lebesgue method needs to compute the measure of sets that are not intervals.
Reference: http://demonstrations.wolfram.com/RiemannVersusLebesgue/
A function (on a compact subset of R^n) is Riemann integrable if and only if it is bounded and continuous almost everywhere. A function defined on the same compact (or on a non compact subset) can be Lebesgue integrable without being bounded. Here the notion of a measurable function is essential. For Lebesgue integrable functions, the integration term by term of sequences of such functions is allowed in conditions which are weaker than that of uniform convergence (the Lebesgue monotone and dominated convergence theorems, Fatou's lemma, etc.). The theory of L^p spaces (1
Dear Dr Alireza Ahmadi
I think the papers
http://scientificbulletin.upm.ro/papers/2011-2/Marginean-Diana-Examples-of-Lebesgue-integrable-functions-which-is-not-Riemann-integrable.pdf
http://math.ucdenver.edu/~jmandel/classes/6131s13/hw10-solution.pdf
https://www.bauer.uh.edu/rsusmel/phd/MR-12.pdf
https://math.berkeley.edu/~arveson/Dvi/105/note1.pdf
and in
https://math.stackexchange.com/questions/1065151/if-a-function-is-riemann-integrable-then-it-is-lebesgue-integrable-and-2-integr
are very useful.
With best Wishes
Dear Dr Alireza Ahmadi
I think the papers
http://scientificbulletin.upm.ro/papers/2011-2/Marginean-Diana-Examples-of-Lebesgue-integrable-functions-which-is-not-Riemann-integrable.pdf
http://math.ucdenver.edu/~jmandel/classes/6131s13/hw10-solution.pdf
https://www.bauer.uh.edu/rsusmel/phd/MR-12.pdf
https://math.berkeley.edu/~arveson/Dvi/105/note1.pdf
and in
https://math.stackexchange.com/questions/1065151/if-a-function-is-riemann-integrable-then-it-is-lebesgue-integrable-and-2-integr
are very useful.
With best Wishes
The class of the Riemann integrable functions are included in the class of Lebesgue integrable functions; i.e. an Riemann integrable function is also Lebesgue integrable. For example, the Dirichlet function is Lebesgue but not Riemann integrable.
The main difference between integrability in the sense of Lebesgue and Riemann is the way we measure 'the area under the curve'.
The Riemann integral asks the question what's the 'height' of ff above a given part of the domain of the function. The Lebesgue integral on the other hand asks, for a given part of the range of ff, what's the measure of the xx's which contribute to this 'height'.
Riemann integral depends on using supremam and infumam for the function f, while, Lebesgue integral depends on the measure for the function at the interval of integral.
See the paper: Jawad: More Differentiability for the Parameter Integral.
Riemann integrability implies Lebesgue integrability but converse not always true for example see Dirichlet function. This function is L- integrable but it is not R- integrable.
following links are useful,
https://mathcs.org/analysis/reals/integ/lebes.html
home.iitk.ac.in/~chavan/afs_ https://www.quora.com/What-is-the-difference-
https://www.quora.com/What-is-the-difference-between-Riemann-integration- and-Lebes...
We know that "A is measurable set iff \chi(A) is a Lebeggue integrablefunction" . Since The Rational numbers set "Q" is a measurable st, so the function \chi(Q) is integrable and its Lebesgue integral is equal to 0.
On the other hand it is easy to see that this function and especially its restriction on the interval [0,1] is not Reiman integrable function, because:
sup U(P,\chi(Q))=1 and inf L(P,\chi(Q))=0.
The Riemann integral on [0,1] for the Dirichlet function is 0, while it is 1 with Lebesgue integral.
Dear Dr Alireza Ahmadi
Greeting
Please see the following links to serve your answer
https://www.quora.com/What-is-the-difference-between-Riemann-integration-and-Lebesgue-integration
https://www.reddit.com/r/askmath/comments/efpdx1/measure_theory_is_there_a_difference_between/
https://planetmath.org/comparisonbetweenlebesgueandriemannintegration
https://mathcs.org/analysis/reals/integ/lebes.html
https://www.jstor.org/stable/44153755?seq=1
https://www.quora.com/What-is-the-difference-between-Riemann-integration-and-Lebesgue-integration
https://math.stackexchange.com/questions/1106636/the-difference-between-riemann-integrable-function-and-lebesgue-integrable-funct?rq=1
https://math.stackexchange.com/questions/1642034/eli5-riemann-integrable-vs-lebesgue-integrable
Best wishes
- Badges
- Science topic
- Similar topics
- Mathematics