Definition of Lebesgue Integral

Question

In class we defined the Lebesgue integral for a nonnegative measurable function $f$ as $ \int_E fd\mu=sup\{\int_E\phi d\mu \mid0< \phi\ <f$ where $\phi$ are simple functions$ \}$. And then we defined that $f$ is Lebesgue integrable if its Lebesgue integral is finite. What bothers me about this definition is what about the "upper Lebesgue Intgrals" i.e What if we consider $inf\{\int_E\phi d\mu \mid0< f<\phi\ $ where $\phi$ are simple functions$ \}$ will this even be finite if the former is? are the two equal? I would expect that the Lebesgue Integrable functions would be those functions where the two definition of Lebesgue integral coincide and are finite.

Can anyone help me understand what is going on?

Thanks in advance

Answer 1

Consider $f(x) = e^{-x}\mathbb 1_{x>0}$. This is a bounded function with exponential decay, so you want to say it has finite integral, but any simple function that satisfies $\phi > f$ will take on a constant positive value on some set of infinite measure.

Even on bounded sets, functions like $\frac1{\sqrt x} \in L^1([0,1])$ cause a problem. Here, there must be a set of positive measure for which $\phi$ takes the value $\infty$.

Answer 2

If $f$ is a bounded function on a set $E$ of finite measure, then both upper and lower Lebesgue integrals are finite.

Given any simple functions $\phi$ and $\psi$ (which by definition are bounded) on $E$ such that $\phi \leqslant f \leqslant\psi$, their Lebesgue integrals always exist and by monotonicity of integrals for simple functions we have

$$\int_E \phi \leqslant \int_E \psi$$

It follows that the supremum (lower integral) and infimum (upper integral) of these integrals are finite with

$$\sup_{\phi \leqslant f} \int_E \phi \leqslant \inf_{\psi \geqslant f}\int_E \psi$$

When the lower and upper integrals are equal, $f$ is said to be Lebesgue integrable where the integral is the common value.

The upper integral is discarded in defining the Lebesgue integral of a nonnegative, measurable, and potentially unbounded function $f$ on a potentially infinite-measure set $E$. However, both upper and lower integrals play a role in defining the Lebesgue integral of any bounded measurable function $g$ with finite support where $0 \leqslant g \leqslant f$. This is used to construct the integral of $f$ as

$$\int_E f = \sup \left\{\int_E g \, |\, g \text{ bounded, measurable, of finite support and } 0 \leqslant g \leqslant f \right\}.$$

Alternatively, you can use your definition in terms of simple functions $\phi \leqslant f$.