## 7.3. Measures

### Example 7.3.1(a): Oddities of Riemann Integral

It depends. For a function to be Riemann integrable it must be bounded. If the function was unbounded even at a single point in an interval*[a, b]*it is not Riemann integrable (because the

*sup*or

*inf*over the subinterval that includes the unbounded value is infinite). Therefore:

That's odd: either a change at a single point should

- If we change the value of a Riemann integrable function to another
boundedvalue at a single point, the Riemann integral would not change at all (prove it).- If we change a the value of a Riemann integrable function to infinity at a single point, then the function is no longer Riemann integrable.

*always*matter, or it should

*never*matter, regardless of the changed value.