Prove Remark 10.5, i.e. prove the linearity of the integral. Data from remark 10.5 If u(x) =
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Prove Remark 10.5, i.e. prove the linearity of the integral.
Data from remark 10.5
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If u(x) = v(x) is defined in R for all x EX-i.e. if we can exclude '-' - then Theorem 10.4(i), (ii) just say that the integral is linear: [(au+Bv)d= [ud + B [vd, , BER. (10.2) This is always true for real-valued u, ve L (), i.e. L (u) is a vector space with addition and scalar (R) multiplication defined by (u + v) (x):= u(x) +v(x), (a.u)(x):= a.u(x), and [... du: L ()R, is a positive linear functional. U+ udp,
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