Royden - section 4.2, page 73 - linearity In Royden's "Real Analysis" on page 73, after the proof of linearity and monotonicity of the Lebesgue integral of simple functions, there's a little paragraph that says that this linearity shows that the disjointness assumption in Lemma 1 (on page 72) is unnecessary. Now I understand why disjointness is unnecessary and linearity but not how they are related, i.e. how linearity shows this. Can anyone shed light on this? Thank you!

Lemma 1 states that if $E_1,E_2,\ldots,E_n$ are disjoint measurable subsets of a finite-measure set $E$, then for a simple function $\phi = \sum_ia_i \cdot \chi_{E_i}$ it follows that

$$\int_E \phi = \sum_{i=1}^n a_i \cdot m(E_i).$$

The proof invokes the definition of the integral of a simple function in terms of the canonical representation, $\phi = \sum_i\lambda_i \cdot \chi_{A_i},$ where the $\lambda_i$'s are distinct and the $A_i$'s are disjoint. The assumption that the $E_i$'s are disjoint is needed because the proof relies on the fact that each set $A_i$ is a disjoint union with measure

$$m(A_i) = \sum_j m(E_{i_j}).$$

However, once you prove linearity of the integral, in general, the conclusion of the lemma is immediate regardless of whether the $E_i$'s are disjoint or not, i.e.

$$\int_E \phi = \int_E\sum_{i=1}^n a_i \cdot \chi_{E_i}= \sum_{i=1}^n a_i \cdot \int_E \chi_{E_i}= \sum_{i=1}^n a_i \cdot m(E_i).$$