I'm seeking interesting examples that are not mentioned in usual real analysis texts. It seems that in general there is no relationship between Riemann integrability and Lebesgue integrability when the function is unbounded or the integral is over an unbounded interval. For example, $\frac{\sin x}{x}$ and $(-1)^n\frac{1}{n}1_{(n,n+1)}$ are Riemann integrable (RI for short) but not Lebesgue integrable (LI for short) over $(0,\infty)$. We can modify the second example to make an unbounded function on $[0,1]$ which is RI but not LI.
We also know that $f$ is LI iff $|f|$ is LI. However, this is not true when it comes to RI (e.g. $f(x)=-1$ or $1$ depending whether $x$ is rational or not; another example is $\frac{\sin x}{x}$). Now consider the following example pattern:
$f$ is/isn't RI AND $|f|$ is/isn't RI AND $f$ is/isn't LI.
Basically we impose three properties to $f$ and for each property we can assign "is" or "is not". So there are 8 possibilities. I myself am able to find examples for five of them. But I can't find the following three:
- Is there a function $f$ such that $f$ is RI and $|f|$ is RI and $f$ is not LI?
- Is there a function $f$ such that $f$ is not RI and $|f|$ is RI and $f$ is not LI?
- Is there a function $f$ such that $f$ is RI and $|f|$ is not RI and $f$ is LI?
Of course all the integrals are taken over the same single interval (most likely unbounded).
These counter examples are very interesting and can deepen our understanding of RI and LI. I'm also seeking more counter example such as:
- Is there a function $h$ such that $h$ is Riemann and Lebesgue integrable but their values are unequal?
- Is there a function $h$ such that $|h|$ is Riemann and Lebesgue integrable but their values are unequal?
Riemann integrability implies Lebesgue integrability, in which case the values of the integrals are equal, taking care of 1, 3, and 4. Since the absolute value function is continuous, and the composition of a continuous function with a Riemann integrable function is Riemann integrable, 2 is false.