# Example of Lebesgue Integral but not Riemann Integrable

1. Nov 16, 2008

### Nusc

What's Example of Lebesgue Integrable function which is not Riemann Integrable?

2. Nov 16, 2008

### morphism

There are plenty. Can you think of a characteristic function of a nonempty measurable set (of finite measure) that is discontinuous everywhere? Why will this do?

3. Nov 16, 2008

### Nusc

There's example 7.4 on page 145 -- in the limit, this is the classic example of a non-Riemann-integrable function.

But I don't understand why this will do.

4. Nov 17, 2008

### HallsofIvy

Staff Emeritus
I get annoyed when people refer to examples in specific books- do they expect everyone to have the book in front of them? But here, you don't even say what book!

The simplest example of a Lebesque integrable function that is not Riemann integrable is f(x)= 1 if x is irrational, 0 if x is rational.

It is trivially Lebesque integrable: the set of rational numbers is countable, so has measure 0. f = 1 almost everywhere so is Lebesque integrable and its integral, from 0 to 1, is 1.

But if no matter how we divide the interval from x= 0 to x= 1 into intervals, every interval contains both rational and irrational numbers: the "lower sum" is always 0 and the "upper sum" is always 1. As we increase the number of intervals to infinity, those do NOT converge.

5. Dec 2, 2008

### vigvig

Consider the following set $A= Q \cap [0,1]$. Where Q is the set of rational numbers of course. Now consider the characteristic function of $A$ denoted $X_A$ defined as follow: $X_A(x)=0$ when $x \in A$ and $x=0$ otherwise. Since this function is almost zero everywhere, then its Lebesgue integral is clearly 0. 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.

Vignon S. Oussa

Last edited: Dec 3, 2008
6. Nov 19, 2011

### riesling

Could you give us another, more complicated example? It seems like the Dirichlet function is everywhere!

thanks!
riesling

7. Nov 19, 2011

### micromass

Staff Emeritus
What kind of example do you want?? You can also have

$$f(x)=x~\text{for}~x\neq \mathbb{Q}~\text{and}~f(x)=0~\text{otherwise}$$

8. Nov 20, 2011

### riesling

Thanks! I'm looking for some where the set of discontinuities and the set of continuities are both of non-zero measure...Is that posible...I know of a type of Cantor set which has positive measure...are there others?