For non-negative functions, the definition of $\int f$ is consistent among Stein, Folland and Wikipedia. However, for real-valued functions, there's a difference.
Stein (page 64) first defines integrable functions, and then says:
If $f$ is Lebesgue integrable, we give a meaning to its integral as follows..."
This implies that whenever we write $\int f$, we are implicitly assuming that both $\int f_+$ and $\int f_-$ are finite.
However, in Wikipedia and Folland, the integral is defined as long as one of $\int f_+$ and $\int f_-$ is finite.
I wonder what impact this discrepancy has on the results about integration.
I think Stein's definition is more convenient for stating results, since we're imposing stronger assumptions on the functions.
However, it seems that there're at least two bad things:
- It's rather limited, as we cannot even define $\int f$ when $f(x)=-1$ on a small interval and $\max\{0,x\}$ elsewhere.
- For a non-negative function $g$, $\int g$ could be $\infty$ by the definition of integral of non-negative functions. But we can also view $g$ as a general real-valued function. Then $\int g$ has to be finite. This seems inconsistent to me.
In my real analysis course, the instructor adopted Stein's definition. I wonder if we adopted Folland's definion, to what extent are all those theorems/lemmas still correct?
I'm a beginner in real analysis. Sorry if this question seems trivial.
The definition of real-valued $f$ being integrable usually means $\int f_+$ and $\int f_-$ are both finite (in other words $\int |f|<\infty$). It's possible to assign values to $\int f$ as long as at least one of $\int f_+,\int f_-$ is finite, but I think it's fair to say that the case of integrability (i.e., absolute integrability $\int|f|<\infty$) is the most important.
When exactly one of $\int f_+,\int f_-$ is infinite, we usually study such an $f$ by considering its vertical truncations to level $n$, $f_n = f\chi_{\{|f|<n\}}$, or its "horizontal" truncations $f_n = f\chi_{[-n,n]^d}$ in $\mathbb R^d$ (or both when possible).
Appropriately truncated functions are absolutely integrable, and results for functions having one of $\int f_+,\int f_-$ infinite can usually be produced straightforwardly as corollaries of what happens in the integrable case by applying the various limit theorems of Lebesgue integration (monotone, dominated, etc.) to their truncations.