Horizontal truncation property of Lebesgue integral From here:

Corollary 11 (Finite additivity of the Lebesgue integral) Let $f, g:
 {\bf R}^d \rightarrow [0,+\infty]$ be measurable. Then $\int_{{\bf
 R}^d} f(x)+g(x)\ dx = \int_{{\bf R}^d} f(x)\ dx + \int_{{\bf R}^d}
 g(x)\ dx$.
Proof: From the horizontal truncation property and a limiting
  argument, we may assume that $f, g$ are bounded[...]

Could anyone tell me why do we assume that $f,g$ are bounded functions? We don't know if they are bounded or not - it's not stated in the assumptions. I just don't see where this conclusion comes from. Doesn't it contradict our assumptions?
Actually what I'm looking for is an informal explanation of what horizontal truncation property is all about, and also why we can assume $f,g$ are bounded, even if they may be unbounded.
 A: Here is a full version of my comment:
The author wants you to consider $f_N = f \cdot 1_{|f| \leq N}$ and the analogous $g_N$. These are bounded, so that (if we have the result for bounded functions) the result holds for these. Then, we pass to the limit $N \to \infty$ and somehow have to verify that $\int f_N + g_N$ converges to $\int f+g$ and that $\int f_N \to \int f$ and likewise for $g$. One possibility here is to use monotone convergence (if that is already known).
Such an argumentation appears frequently in mathematical proofs. If the author just writes "we may assume [without loss of generality] that", it is meant that one can modify the original problem in a (more or less) obvious way so that the additional assumption is satisfied.
Afterwards, one can then derive the result for the original objects (functions, ...) by that for the modified ones, e.g. by taking some sort of limit, using linearity, ...
If the author thinks that these steps are obvious, they are (sadly) often omitted.
Otherwise, frequently at some point in the proof the modification and justification (e.g. for taking the limit) are spelled out explicitly.
It often happens to me that I get frustrated by not seeing the "obvious" modifications; then I stop reading to figure it out. Only after a lot of thinking, I then sometimes realize that the author tells me how to do it later in the proof :)
