"It is easy to see...."

The integral (I) of function f with respect to measure u is defined by

I (f du) = I (f+ du) - I (f- du)

where f+ and f- are the positive and negative parts of f. Now if
f = v - w, where u and v are any non-negative functions, my text
(Bartle) goes on to say that "it is easy to see" that

I (f du) = I (v du) - I (w du)

How is it possible to arrive at this assuming only the definition of
(Lebesque) integral of non-negative functions? Many thanks.

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