Question 8.T.7: (Positivity Property) If f ∈ R(a, b) is non-negative, then a...

(Positivity Property)

If f ∈ \mathcal{R}(a, b) is non-negative, then

\int_{a}^{b}{f ≥ 0}.

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Let P = \left\{x_{0}, x_{1}, …, x_{n}\right\} be any partition of [a, b]. Since f ≥ 0 on [a, b] , m_{i} ≥ 0 for all i ∈ {0, 1, . . . , n − 1} and therefore L (f, P ) ≥ 0.

Now

\int_{a}^{b}{f }= L (f) = \sup \left\{L (f, P) : P ∈ \mathcal{P} (a, b)\right\} ≥ 0.

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