Algebraic Geometry: A New Treatise On Analytical Conic by William Martin Baker

By William Martin Baker

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Images of sets under functions are not as well behaved as inverse images, nonetheless we do have the following result—which is easily verified. 6. 10 Let f : X → Y . (1) If {Aα}α∈I is a collection of subsets of X, then f ( (2) If X ⊇ A1 ⊇ A2 ⊇ · · · , then f ( inclusion is possible. ∞ i=1 Ai) ⊆ ∞ i=1 α∈I Aα) = α∈I f (Aα ). f (Ai ) holds and strict To obtain an equality for images of intersections, we need to look at continuous functions and decreasing sequences of compact sets. 11 Let X and Y be topological spaces and let f : X → Y be continuous.

But then the inequality φδ (I) ≥ φδ (I1) + φδ (I2) clearly fails. It is not difficult to show that, if all members of F are Borel sets, then every subset A of RN is contained in a Borel set A with the same φδ measure (just take the intersection of the unions of covers). Thus ψ is a Borel regular measure. We now describe an alternative approach to Carath´eodory’s construction that is due to Federer [Fed 54]. In fact ψ(A) can be characterized as the infimum of the set of all numbers t with this property: For each open covering U of A there exists a countable subfamily G of F such that each member of G is contained in some member of U , G covers A, and ζ(S) < t .

1 The Basic Definition Let F be a collection of sets in RN . These will be our “test sets” for constructing Hausdorff-type measures. Let ζ : F → [0, +∞] be a function (called the gauge of the measure to be constructed). 1: Carath´eodory’s construction. 1)    . 1). 1(1) and thus is a measure. If 0 < δ1 < δ2 ≤ ∞ then it is immediate that φδ1 ≥ φδ2 . Thus we may set ψ(A) = lim+ φδ (A) = sup φδ (A) . δ→0 δ>0 Certainly ψ is also a measure. This process for constructing the measure ψ is called Carath´eodory’s construction.

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