※ 引述《kemowu (小展)》之銘言：
: Suppose ∫ f(y) dy = 0 for every subinterval I of R. Show that f = 0 a.e.
: I
: on R.
(i) From the assumption, μ(E):= ∫f is a finite signed measure on the
E
measurable space (R,M), M is the colleciton of all (Leb.)measurable set
in R.
(ii) Consider that any open sets is the union of some disjoint open
intervals. Hence we can show that μ(B)= 0 for any open sets B by
the additive property for the measure μ.
(iii)For any measurable set E, μ(E)=μ(∩Bi) for some open set Bi,
(In fact, E=(∩Bi)\Z, Z is a null set). WLOG, we may assume Bi is
decreasing, so μ(E)=μ(∩Bi) = limμ(Bi) = 0 by (ii).
Using the fact above, we can obtain the desired result.
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