## Example 8.2.9: Convergence Almost Everywhere

Let *r _{n}* be the (countable) set of rational numbers
inside the interval

*[0, 1]*, ordered in some way, and define the functions

Show the following:and

- The sequence
*g*converges pointwise to_{n}*g*but the sequence of Riemann integrals of*g*does not converge to the Riemann integral of_{n}*g*. - The sequence
*g*converges a.e. to zero and so does the sequence of Lebesgue integrals of_{n}*g*._{n}

Each *g _{n}* is continuous except for finitely many points
of discontinuity. But then each

*g*is integrable and it is easy to see that

_{n}g_{n}(x) dx = 0

But the limit function is not Riemann-integrable and hence the sequence of Riemann integrals does not converge to the Riemann integral of the limit function.

Please note that while each *g _{n}* is continuous
except for finitely many points, the limit function

*g*is discontinuous

*everywhere*

On the other hand, each *g _{n}* is zero except on a set
of measure zero, and so is the limit function . Thus, using
Lebesgue integration we have that all integrals evaluate to zero. But then, in
particular, the sequence of Lebesgue integrals of

*g*converge to the Lebesgue integral of

_{n}*g*.