Let a<b be real numbers and let h:[a,b]→R be continuous on the closed interval [a,b]. Define the zero extension h~:R→R by h~(x)=h(x) for x∈[a,b] and h~(x)=0 otherwise. Then the following hold. Claim 1. h is…
Let (Xm)m∈N and X be random variables, not necessarily on a common probability space, and let R denote the real numbers. Call a function f:R→R an admissible test function if f is bounded, f is a C3 map on…
Let R denote the real numbers and let f:R→R be a C3 map on R=R1, and suppose its third derivative is bounded: there is M3≥0 with ∣f′′′(x)∣≤M3 for all x∈R, where f′, f′′, f′′′ denote the iterat…
Let g(x)=exp(−x2/2) with the exponential function, let λ be Lebesgue measure, and for a Borel set B let ν(B)=∫R1Bgdλ, the integral of Lebesgue Integral of a Nonnegative Measurable Function of the measurable function…
By claim 5 of Basic Properties of the Exponential Function, the exponential function is a bijection from R onto the interval (0,∞). The natural logarithm is its inverse function…
Let exp be the exponential function. Then: 1. exp(0)=1 and exp(u+v)=exp(u)exp(v) for all u,v∈R; 2. exp(u)>0 for every u∈R, and exp(−u)=1/exp(u); 3. exp is differentiable at every point with exp′=exp, where the derivative is the…
The exponential function exp:R→R is defined by exp(u)=∑k=0∞k!uk, with factorials and the convention u0=1. The series converges for every u∈R: for indices k>2∣u∣ the terms are dominated in absolute valu…
Let (X,F,μ) and (Y,G,ν) be σ-finite measure spaces and let μ⊗ν be the product measure on the product σ-algebra. Sections. For f:X×Y→[0,∞] measurable with respect to F⊗G (in the sense…
Let (X,F,μ) and (Y,G,ν) be measure spaces, and suppose that both μ and ν are σ-finite. Then there exists exactly one measure μ⊗ν on the product σ-algebra F⊗G such that…
Let X be a set. A family P of subsets of X is a π-system if it is nonempty and closed under finite intersections: A,B∈P implies A∩B∈P. A family L of subsets of X is a λ-system if: (1) X∈L; (2…
Let (X,F) and (Y,G) be measurable spaces. A measurable rectangle is a subset of the Cartesian product X×Y of the form A×B with A∈F and B∈G. The product σ-algebra F⊗G on…
Let (X,F,μ) be a measure space. 1. (Nonnegative case.) Let f,g:X→[0,∞] be measurable and let c∈[0,∞). Then f+g and cf are measurable, and ∫X(f+g)dμ=∫Xfdμ+∫Xgdμ,∫Xcfdμ=c∫Xfdμ, with the con…
Let (X,F,μ) be a measure space, and let (fm)m∈N be a sequence of measurable functions fm:X→R such that for every x∈X the sequence (fm(x))m converges to f(x), for a function f:X→R. Suppose there is an integrable…
Let (X,F,μ) be a measure space and let (fm)m∈N be a sequence of measurable functions fm:X→[0,∞]. For x∈X define (liminfmfm)(x)=supk∈Ninfm≥kfm(x), where the infimum and supremum are taken i…
Let (X,F,μ) be a measure space and let (fm)m∈N be a sequence of measurable functions fm:X→[0,∞] such that fm(x)≤fm+1(x) for every x∈X and every m. Define f:X→[0,∞] pointwise by f(x)=supmfm(x), the…
Let (X,F,μ) be a measure space and let f:X→R be measurable. Define the positive part f+ and negative part f− by f+(x)=max{f(x),0},f−(x)=max{−f(x),0}, so that f=f+−f− and ∣f∣=f++f−. Both f+ and…
Let (X,F,μ) be a measure space. A function f:X→[0,∞] (values in the extended half-line of Measure, Measure Space, and Probability Measure) is called measurable if {x∈X:f(x)>a}∈F for every a∈R; for real-valued f this ag…
Let (X,F,μ) be a measure space. For A⊆X, the indicator function 1A:X→R is defined by 1A(x)=1 for x∈A and 1A(x)=0 otherwise. A simple function on (X,F) is a measurable function…
Let (X,F) and (Y,G) be measurable spaces. A function f:X→Y is measurable (with respect to F and G) if f−1(B)∈F for every B∈G. A real-valued function f:X→R is called measurable if it…
Let λ∗ be the Lebesgue outer measure on the real line. Then: 1. λ∗ is an outer measure on R; 2. every Borel set is Carathéodory measurable with respect to λ∗; 3. consequently, by Caratheodory Extension Theorem, the restriction…