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Let r ∈ Fin( ∗R). Then there is a unique s ∈ R such that r ≈ s. Proof. Define s := sup{u ∈ R | u ≤ r}. By r being finite and the Monotone Convergence Theorem, s ∈ R. If s ≈ r, let ∈ R such that 0 < < |r − s| . Then either s + < r or s − > r, both cases contradict to the definition of s. Now let s1 , s2 ∈ R and s1 ≈ r ≈ s2 , then s1 ≈ s2 , so s1 − s2 = 0, the only infinitesimal in R. By the proposition, the following is well-defined: ◦ : Fin( ∗R) → R, where ◦ r ∈ R and r ≈ ◦ r, the standard part of r.

Nonstandard Analysis 7 This is known as the Axiom of Choice (AC). Among all axioms of ZFC, it may look like an unusual one. As a matter of fact, until quite recently most mathematicians were suspicious of AC for its nonconstructive nature. However, AC is accepted nowadays for pragmatic reasons such as the existence of cardinality or the claim that every vector space has a basis. AC is independent of ZF—the rest of ZFC. That is, ZF AC and ZF ¬AC. Another famous result is that CH, the Continuum Hypothesis, the assertion that |R| = ℵ1 , is independent of ZFC.

13. Let Ω, B, µ be a finite internal complex measure space. Then there are finite internal positive measures µ11 , µ12 , µ21 , µ22 such that µ ≈ (µ11 − µ12 ) + i(µ21 − µ22 ). Moreover, the decomposition ◦ µ = ( ◦ µ11 − ◦ µ12 ) + i( ◦ µ21 − ◦ µ22 ) is unique. Proof. Let µ1 := Re(µ) and µ2 := Im(µ), so µ1 , µ2 are finite internal real-valued measures. Since µ1 is finite, for some r ∈ Fin( ∗R), µ1 : B → [−r, r]. By saturation, there is A ∈ B such that µ1 (A) ≈ inf X∈B µ1 (X) ≤ 0. Now define finite internal positive measures µ11 , µ12 : Ω → [0, r] by µ11 (X) = µ1 (X \ A) and µ22 (X) = −µ1 (X ∩ A), X ∈ B.