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Extra resources for Lectures on complex analytic varieties: The local parametrization theorem
Sample text
3. Structure of measurable functions 15 We could use the (discontinuous9 ) convention 0 · ∞ = 0 and10 it is easy to verify that this new multiplication is associative, commutative, with neutral element 1, distributive with respect to the addition. 4 below. 1. Let (X, M) be a measurable space and let (fn )n∈N be a sequence of measurable functions from Xinto R. Then the functions sup fn , inf fn , lim sup fn , lim inf fn are measurable. In particular, the pointwise limit of a sequence of measurable functions is measurable.
N − 1, n} so that μ(A) = k k n−k . k∈A Cn p (1 − p) (13) The Poisson probability with parameter λ > 0 is given by e−λ k∈N λk δk , k! 2. We may consider μ as defined on the k subsets of N by μ(A) = e−λ k∈A λk! 3. Let (X, M, μ) be a measure space where μ is a positive measure and let f : X −→ Y be a mapping. The set N = {B ⊂ Y, f −1 (B) ∈ M} is a σalgebra on Y : it is the largest σ-algebra on Y making f measurable. The so-called pushforward measure f∗ (μ) is a positive measure defined on N by f∗ (μ)(B) = μ f −1 (B) .
1(1). , then X gdμ. 1. 4. Let (X, M, μ) be a measure space where μ is a positive measure. (1) The mapping from L1 (μ) into C defined by f → X f dμ is a linear form. (2) The mapping from L1 (μ) into R+ defined by f → X |f |dμ = f L1(μ) is a norm and for f ∈ L1 (μ) f dμ ≤ X |f |dμ. B. 2 in Chapter 2 the introduction of spaces Lp (μ) along with the proof that these spaces are complete. Proof. 3, and for the same reason, the mapping defined in (2) makes sense on the quotient space L1 (μ). , f = 0 in L1 (μ).