TITLE: Sum of measures and integral
QUESTION [24 upvotes]: Suppose we have a measure that can be expressed as linear combination of 2 measures: $m=am_1+bm_2$. What does this imply for the calculation of the integral? Do we have:
$$\int f\,dm=a\int f\,dm_1 + b\int f\,dm_2$$
Here I consider a positive measure with $f$ some measurable map. The example I have in mind is the sum of dirac measures $1/2(\delta_0+\delta_1)$ on $[0,1]$.

REPLY [17 votes]: This works fine when all the measures are finite, or $a$ and $b$ are non-negative. 
We can check the mentioned formula considering first simple functions (linear combinations of characteristic functions of measurable sets), then jump to the general case approximating almost everywhere by a sequence of simple functions.