If f is integrable on [a, b] and c ? , then cf is integrable on [a, b] and

(15.12)

If c=0 the theorem is obvious. Let c ?0, ?>0. There exists a positive function ? such that

whenever TX ? ?. It follows that

If cf is integrable and c ?0 then f is also integrable and Equation (15.12) holds.

If f and g are integrable on [a, b] then so is f+g and

(15.13)

For every positive ? there exist positive functions ? ₁ and ? ₂ such that

(15.14)

for TX ? ? ₁, and

(15.15)

for TX ? ? ₂ . Define ?(x)=Min( ? ₁ (x), ? ₂ (x)). If TX ? ? then Inequalities (15.14) and (15.15) hold and since R(f+g, TX)=R(f, TX)+R(g, TX), we have

It is an easy exercise to extend Formula (15.13) to a sum of n functions.

If S is countable and f(x)=0 for x ? [a, b]\S then f is integrable and

Obviously we can assume that S is enumerable, (otherwise we enlarge it by joining it with an arbitrary enumerable part of [a, b]). Let n