Linearity of integration for nonnegative measurable functions on pg.80 Royden "4th edition".

Question

The statement of the theorem is given below:

But I do not know why the assumption $\alpha > 0$ and $\beta >0,$ could anyone explain this for me, please?

Answer 1

Because Section 4.3 of Royden, where you see Theorem 10, is "the Lebesgue integral of a measurable nonnegative function". The authors have not yet defined the general Lebesgue integral (see Section 4).

In the proof, the authors want to talk about $\int \alpha f$ and $\int \beta g$, which only makes sense for $\alpha\ge 0$ and $\beta\ge 0$ at this point.

Answer 2

The integrals of $f,g$ may by infinite.

So you need $a,b>0$ if $f,g$ are non-negative,to avoid cases of $\infty-\infty$