Could you give me concrete examples about
"a function that is bounded and measurable but not Lebesgue integrable".
Royden's textbook "Real analysis" says a bounded measurable function is said to be integrable if its lower Lebesgue integrale is equal to its upper Lebesgue integral.
(I know if the domain is of finite measure, then a bounded function is Lebesgue integrable iff it is measurable, so my desired example need to be on a domain of infinite measure.)
Best Answer
Let $f : \Bbb R \to \Bbb R$ be defined as:
$$f(x) = \begin{cases} 1 & x \in [0, \infty) \\ 0 & \text{else} \end{cases}.$$ Clearly, $f$ is measurable since $f = \chi_{[0, \infty)}$ (and $[0, \infty)$ is a Lebesgue measurable set, so its characteristic function is measurable).
Also clearly $f$ is bounded. But $\int \limits_{\Bbb R} |f| \,dm = \infty$.