A basic doubt on integration

Question

I have seen in books that integral is defined separately for positive and negative parts and then subtraction gives actual integral. What if we have done it without dividing into positive and negative part. For example in case of Riemann integral. This is not clear to me.

Answer 1

Edit: Maybe you would like to check out this question at MathOverflow. They discuss some differences between the approaches which use and don't use the positive/negative parts. As the question and one of the comments in the accepted answer remark, Lang's Real and Functional analysis and Halmos' Measure theory books treat the Lebesgue integral without the positive/negative part approach. Both approaches have their own advantages.

I can only talk to you about the one I learned, the positive/negative part approach, which I personally find tidy. I also like that it almost immediately presents results such as the monotone convergence theorem and Fatou's lemma.

The Lebesgue integral for positive functions is defined using the integrals of the simple functions $(\phi_n (x))_{n\geq 0}$ which approximate the function from below in an increasing sequence, that is, $\phi_n(x) \leq \phi_{n+1}(x)$, for all $x$. These simple functions are easily described when the function only takes values of one sign. On the other hand, as the sequence is increasing, you just define the integral as the "lim sup" of the values of the integrals of those simple functions (admitting the value $+\infty$ if that is the case).

You can make the same argument for functions which only take negative values (by considering $-f$ instead of $f$ and defining the integral value as $-\infty$ in the case that the lim sup is $+\infty$), and doing so you define the integral for negative functions too.

There is nowhere a proper limit to be found in these definitions, just lim sups. This is in some way "less to ask" for the functions you are integrating (apart from the fact that there are more measurable functions than piecewise continuous functions).

Now, the integral for general functions can be defined by splitting the function as the difference of its positive and negative parts (which you can prove to be measurable as well) only when at least one of those two integral values is finite, because you can't define $\infty -\infty$ properly. If both are finite, you get a finite answer and that is the integrated value. If one of them is infinite, you get $+\infty$ or $-\infty$ depending of the case.

I'm not aware if you can define Riemann integrals in the same way, considering positive and negative parts, but I think it wouldn't be necessary.