48 1. Measure theory a sequence f1,f2,f3,... : Rd [0, +∞] of unsigned simple functions such that fn(x) f(x) for every x Rd. This particular definition is not always the most tractable. Fortunately, it has many equivalent forms: Lemma 1.3.9 (Equivalent notions of measurability). Let f : Rd [0, +∞] be an unsigned function. Then the following are equivalent: (i) f is unsigned Lebesgue measurable. (ii) f is the pointwise limit of unsigned simple functions fn (thus the limit limn→∞ fn(x) exists and is equal to f(x) for all x Rd). (iii) f is the pointwise almost everywhere limit of unsigned simple func- tions fn (thus the limit limn→∞ fn(x) exists and is equal to f(x) for almost every x Rd). (iv) f is the supremum f(x) = supn fn(x) of an increasing sequence 0 f1 f2 . . . of unsigned simple functions fn, each of which are bounded with finite measure support. (v) For every λ [0, +∞], the set {x Rd : f(x) λ} is Lebesgue measurable. (vi) For every λ [0, +∞], the set {x Rd : f(x) λ} is Lebesgue measurable. (vii) For every λ [0, +∞], the set {x Rd : f(x) λ} is Lebesgue measurable. (viii) For every λ [0, +∞], the set {x Rd : f(x) λ} is Lebesgue measurable. (ix) For every interval I [0, +∞), the set f −1(I) := {x Rd : f(x) I} is Lebesgue measurable. (x) For every (relatively) open set U [0, +∞), the set f −1(U) := {x Rd : f(x) U} is Lebesgue measurable. (xi) For every (relatively) closed set K [0, +∞), the set f −1(K) := {x Rd : f(x) K} is Lebesgue measurable. Proof. (i) and (ii) are equivalent by definition. (ii) clearly implies (iii). As every monotone sequence in [0, +∞] converges, (iv) implies (ii). Now we show that (iii) implies (v). If f is the pointwise almost everywhere limit of fn, then for almost every x Rd one has f(x) = lim n→∞ fn(x) = lim sup n→∞ fn(x) = inf N0 sup n≥N fn(x).
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