Can a function be integrable but not Riemann integrable?

You mean to be Lebesgue integrable and not Riemann integrable? The answer is yes. Classic example, let f(x)=1 if x is a rational number and zero otherwise on the interval [0,1].

What function is not Riemann integrable?

An unbounded function is not Riemann integrable. In the following, “inte- grable” will mean “Riemann integrable, and “integral” will mean “Riemann inte- gral” unless stated explicitly otherwise. f(x) = { 1/x if 0 < x ≤ 1, 0 if x = 0. so the upper Riemann sums of f are not well-defined.

Which functions are Lebesgue integrable?

Basic theorems of the Lebesgue integral If f, g are functions such that f = g almost everywhere, then f is Lebesgue integrable if and only if g is Lebesgue integrable, and the integrals of f and g are the same if they exist. The value of any of the integrals is allowed to be infinite.

What is the difference between Lebesgue and Riemann integral?

What is the difference between Riemann Integral and Lebesgue Integral? The Lebesgue integral is a generalization form of Riemann integral. The Lebesgue integral allows a countable infinity of discontinuities, while Riemann integral allows a finite number of discontinuities.

Are Lebesgue integrable functions Riemann integrable?

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.

Are all Lebesgue integrable functions Riemann integrable?

The Riemann integral is only defined for bounded functions on bounded intervals, which are all Lebesgue-integrable. It’s the extension to the improper Riemann integral that can integrate functions that are not Lebesgue-integrable.

Can a discontinuous function be Riemann integrable?

Discontinuous functions can be integrable, although not all are. Specifically, for Riemann integration (our normal basic notion of integrals) a function must be bounded and defined everywhere on the range of integration and the set of discontinuities on that range must have Lebesgue measure zero.

Why are some functions not integrable?

The simplest examples of non-integrable functions are: in the interval [0, b]; and in any interval containing 0. These are intrinsically not integrable, because the area that their integral would represent is infinite. There are others as well, for which integrability fails because the integrand jumps around too much.

What is the difference between Riemann integral and Lebesgue integral?

The Riemann integral is only defined for bounded functions on bounded intervals, which are all Lebesgue-integrable. It’s the extension to the improper Riemann integral that can integrate functions that are not Lebesgue-integrable.

How do you know if a function is Lebesgue integrable?

Hence, if your function were Lebesgue integrable, its Lebesgue integral would have to be equal to any real number. For r > 0, your function has both a Riemann and a Lebesgue integral over [ 0, r], and they have the same value in terms of r.

Is $X_a$ Riemann integrable?

However, it is easily proved that $X_A$ is not Riemann integrable. As an argument but not proof to support this, the function is discontinuous at uncountable number of points. Could you give us another, more complicated example?