### Generalized Riemann integral

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If all intervals had this property, then this would conclude the proof, because each term in the Riemann sum would be bounded by a corresponding term in the Darboux sums, and we chose the Darboux sums to be near s. Instead, it may stretch across two of the intervals determined by y 0 , In symbols, it may happen that. To handle this case, we will estimate the difference between the Riemann sum and the Darboux sum by subdividing the partition x 0 , To bound the other term, notice that.

Any Riemann sum of f on [0, 1] will have the value 1, therefore the Riemann integral of f on [0, 1] is 1. This function does not have a Riemann integral. To prove this, we will show how to construct tagged partitions whose Riemann sums get arbitrarily close to both zero and one. To start, let x 0 , The t i have already been chosen, and we can't change the value of f at those points. But if we cut the partition into tiny pieces around each t i , we can minimize the effect of the t i.

Our first step is to cut up the partition. Now we add two cuts to the partition for each t i. If one of these leaves the interval [0, 1], then we leave it out. If t i is directly on top of one of the x j , then we let t i be the tag for both intervals:. We still have to choose tags for the other subintervals. We will choose them in two different ways. The first way is to always choose a rational point , so that the Riemann sum is as large as possible.

## The Riemann, Lebesgue and Generalized Riemann Integrals

The second way is to always choose an irrational point, so that the Riemann sum is as small as possible. Since we started from an arbitrary partition and ended up as close as we wanted to either zero or one, it is false to say that we are eventually trapped near some number s , so this function is not Riemann integrable.

However, it is Lebesgue integrable. In the Lebesgue sense its integral is zero, since the function is zero almost everywhere. But this is a fact that is beyond the reach of the Riemann integral. There are even worse examples. I Q is equivalent that is, equal almost everywhere to a Riemann integrable function, but there are non-Riemann integrable bounded functions which are not equivalent to any Riemann integrable function. Therefore, g is not Riemann integrable. It is popular to define the Riemann integral as the Darboux integral. This is because the Darboux integral is technically simpler and because a function is Riemann-integrable if and only if it is Darboux-integrable.

Some calculus books do not use general tagged partitions, but limit themselves to specific types of tagged partitions. If the type of partition is limited too much, some non-integrable functions may appear to be integrable. One popular restriction is the use of "left-hand" and "right-hand" Riemann sums. Alone this restriction does not impose a problem: we can refine any partition in a way that makes it a left-hand or right-hand sum by subdividing it at each t i.

In more formal language, the set of all left-hand Riemann sums and the set of all right-hand Riemann sums is cofinal in the set of all tagged partitions. Another popular restriction is the use of regular subdivisions of an interval.

## The generalized Riemann integral on locally compact spaces

For example, the n th regular subdivision of [0, 1] consists of the intervals. Again, alone this restriction does not impose a problem, but the reasoning required to see this fact is more difficult than in the case of left-hand and right-hand Riemann sums. However, combining these restrictions, so that one uses only left-hand or right-hand Riemann sums on regularly divided intervals, is dangerous.

If a function is known in advance to be Riemann integrable, then this technique will give the correct value of the integral. But under these conditions the indicator function I Q will appear to be integrable on [0, 1] with integral equal to one: Every endpoint of every subinterval will be a rational number, so the function will always be evaluated at rational numbers, and hence it will appear to always equal one.

The problem with this definition becomes apparent when we try to split the integral into two pieces.

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The following equation ought to hold:. If we use regular subdivisions and left-hand or right-hand Riemann sums, then the two terms on the left are equal to zero, since every endpoint except 0 and 1 will be irrational, but as we have seen the term on the right will equal 1.

As defined above, the Riemann integral avoids this problem by refusing to integrate I Q. The Lebesgue integral is defined in such a way that all these integrals are 0. Because the Riemann integral of a function is a number, this makes the Riemann integral a linear functional on the vector space of Riemann-integrable functions. A bounded function on a compact interval [ a , b ] is Riemann integrable if and only if it is continuous almost everywhere the set of its points of discontinuity has measure zero , in the sense of Lebesgue measure. This is known as the Lebesgue's integrability condition or Lebesgue's criterion for Riemann integrability or the Riemann—Lebesgue theorem.

It is due to Lebesgue and uses his measure zero , but makes use of neither Lebesgue's general measure or integral. The integrability condition can be proven in various ways, [4] [5] [6] [7] one of which is sketched below. In particular this is also true for every such finite collection of intervals.

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Thus these intervals have a total length of at least c. Since this is true for every partition, f is not Riemann integrable. Now, suppose that f is continuous almost everywhere. To this end, we construct a partition of [ a , b ] as follows:. These neighborhoods consist of an open cover of the interval, and since the interval is compact there is a finite subcover of them. Thus the partition divides [ a , b ] to two kinds of intervals:. In particular, any set that is at most countable has Lebesgue measure zero, and thus a bounded function on a compact interval with only finitely or countably many discontinuities is Riemann integrable.

An indicator function of a bounded set is Riemann-integrable if and only if the set is Jordan measurable. If a real-valued function is monotone on the interval [ a , b ] it is Riemann-integrable, since its set of discontinuities is at most countable, and therefore of Lebesgue measure zero. If a real-valued function on [ a , b ] is Riemann-integrable, it is Lebesgue-integrable.

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That is, Riemann-integrability is a stronger meaning more difficult to satisfy condition than Lebesgue-integrability. If f n is a uniformly convergent sequence on [ a , b ] with limit f , then Riemann integrability of all f n implies Riemann integrability of f , and. However, the Lebesgue monotone convergence theorem on a monotone pointwise limit does not hold.

In Riemann integration, taking limits under the integral sign is far more difficult to logically justify than in Lebesgue integration. It is easy to extend the Riemann integral to functions with values in the Euclidean vector space R n for any n. In particular, since the complex numbers are a real vector space , this allows the integration of complex valued functions.

The Riemann integral is only defined on bounded intervals, and it does not extend well to unbounded intervals. The simplest possible extension is to define such an integral as a limit, in other words, as an improper integral :. This definition carries with it some subtleties, such as the fact that it is not always equivalent to compute the Cauchy principal value.

By symmetry,. But there are many ways for the interval of integration to expand to fill the real line, and other ways can produce different results; in other words, the multivariate limit does not always exist. We can compute. In general, this improper Riemann integral is undefined. Even standardizing a way for the interval to approach the real line does not work because it leads to disturbingly counterintuitive results. If we agree for instance that the improper integral should always be.

Unfortunately, the improper Riemann integral is not powerful enough. The most severe problem is that there are no widely applicable theorems for commuting improper Riemann integrals with limits of functions. In applications such as Fourier series it is important to be able to approximate the integral of a function using integrals of approximations to the function. For proper Riemann integrals, a standard theorem states that if f n is a sequence of functions that converge uniformly to f on a compact set [ a , b ] , then.

On non-compact intervals such as the real line, this is false. For all n we have:. This demonstrates that for integrals on unbounded intervals, uniform convergence of a function is not strong enough to allow passing a limit through an integral sign. Definition 2. Remark 2. In Definition 2. We next give new definition, to be referred as generalized beta r , g -preinvex function. Also, for and we get the notion of generalized s, m -preinvex function see 3.

In this section, in order to prove our main results regarding some new integral inequalities involving generalized beta r , g -preinvex functions, we need the following new Lemma:. Lemma 2. Theorem 2. Since is a generalized beta r , g -preinvex function on K , combining with Lemma 2. Corollary 2.

Under the same conditions as in Theorem 2. Since f l is a generalized beta r, g -preinvex function on K, combining with Lemma 2. Hermite-Hadamard type fractional integral inequalities for generalized beta r, g -preinvex functions.

In this section, we prove our main results regarding some generalizations of Hermite-Hadamard type inequalities for generalized beta r, g -preinvex functions via fractional integrals. Theorem 3. Corollary 3. Under the same conditions as in Theorem 3. Remark 3. For different choices of positive values etc. Kashuri and R. Akkurt and H. Yildirim, On some fractional integral inequalities of Hermite-Hadamard type for r-preinvex functions, Khayyam J. Du, J.

Liao and Y. Li, Properties and integral inequalities of Hadamard-Simpson type for the generalized s, m -preinvex functions, J. Nonlinear Sci.

## An invariant riemann type integral defined by figures — Penn State

Dragomir, J. Persson, Some inequalities of Hadamard type, Soochow J. Hudzik and L. Maligranda, Some remarks on s-convex functions, Aequationes Math. Antczak, Mean value in invexity analysis, Nonlinear Anal. Yang, X. Yang and K. Teo, Generalized invexity and generalized invariant monotonicity, J. Theory Appl. Pini, Invexity and generalized convexity, Optimization. Kavurmaci, M. Avci and M.

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Chu, G. Wang and X. Zhang, Schur convexity and Hadamard's inequality, Math. Zhang, Y. Chu and X. Chu, M. Khan, T. Khan and T. Adil Khan, Y. Khurshid, T. Ali and N.