Integration Flashcards
Definie addition by infinity
For all real x, x + ∞ = ∞ + x = ∞
Define subtraction by infinity
For all real x, x − ∞ = −∞ + x = −∞,
Define multiplication by infinity
For all real x, x.∞ = ∞.x = (−x).(−∞) and x.∞ = ∞ for x>0, x.∞ = -∞ for x<0 and x.∞ = 0 for x=0.
Define the limsup of a sequence
If a_n is a sequence, let b_m = sup(a_n), for all n≥m, then the limsup of a_n is the limit of b_m.
Define the outer measure
m*(A) = inf(The sum of |I_n|, where I_n are intervals whose union contains A).
Define a null set
A set A is null if m*(A) = 0.
Define the Cantor set
Let C0 = [0,1], C1 = [0,1/3] ∪ [2/3,1], C2 = [0,1/9]∪ [2/9,1/3] ∪ [2/3, 7/9] ∪ [8/9, 1], etc. Then the Cantor set, C = the infinite intersection of Cn.
Define almost everywhere
A property hold almost everywhere if the set A not satisfying the property is null.
Define Lebesgue measurable
A subset E of R is Lebesgue measurable if m(A) = m(A∩E) + m*(A\E).
Define a σ-algebra
Let Ω be any set and F ⊆ P(Ω). Then, F is a σ-algebra on Ω if:
1. ∅ ∈ F
2. If E ∈ F, then Ω\E ∈ F
3. If En ∈ F for n = 1, 2,…, then the infinite union of En ∈ F.
Define a measure
A measure on (Ω, F) is a function µ: F → [0,∞] such that:
1. µ(∅) = 0
2. µ(infinite union of En) = Infinite sum of µ(En) whenever En are disjoint sets in F.
Define a measure space
A triple (Ω, F, µ) is a measure space.
Define a finite measure
A measure µ is finite if µ(Ω) < ∞.
Define a probability measure
A measure µ is a probability measure if µ(Ω) = 1.
Define the Borel measure
The smallest σ-algebra containing all the intervals.
Define a measurable function
f is F-measurable if and only if f^(−1)G ∈ F for all G in the measure space.
Define a simple function
A function which takes a finite number of values.
Define the standard form of a simple function
Let φ be a simple function. If it is written as the sum of (α_i)(χ_Bi), where αi are distinct and non-zero and Bi are disjoint, then φ is in standard form.
Define the integral of a non-negative simple function
For a non-negative simple function φ with standard form given by the sum of (α_i)(χ_Bi), the integral of φ is defined to be the sum of (α_i)m(Bi).
Define the integral of a non-negative function
Given a non-negative function f,the integral of f over a measurable set E = sup{integral over R of φ simple, where 0 ≤ φ ≤ f on E and is 0 on R\E}.
Define integrable
A function is integrable over a set if the integral over that set is finite.
State the monotone convergence theorem
If (fn) is an increasing sequence of non-negative measurable functions and f = lim(n→∞)fn, then the integral of f = lim(n→∞) of the integral of fn.
State the baby monotone convergence theorem
Let f be a non-negative measurable function, (En) be an increasing sequence of measurable sets, and E = infinite union of En. Then the integral of f over E = lim(n→∞) of the integral of f over En f.
State the comparison test
If f is measurable and |f| ≤ g for some integrable function g, then f is integrable. If |f| ≥ g ≥ 0 for some measurable function g which is not integrable, then f is not integrable.
State the fundamental theorem of calculus
Let g be a function with a continuous derivative on a closed bounded interval [a, b]. Then g’ is integrable over [a, b], and the integral from a to b of g’(x) = g(b) − g(a).
State integration by parts
Let f and g be continuously differentiable functions on a closed bounded interval [a, b]. Then the integral from a to b of f(x)g’(x) = f(b)g(b) − f(a)g(a) – integral from a to b of f’(x)g(x).
State integration by substitution
Let g: I → R be a monotonic function with a continuous derivative on an interval I, and let J be the interval g(I). A measurable function f: J → R is integrable over J if and only if (f◦g).g’ is integrable over I. Then the integral of f(x) over J = integral of f(g(y))|g’(y)| over I.
State Fatou’s lemma
Let (fn) be a sequence of non-negative measurable functions. Then the integral of liminf(fn) ≤ liminf(the integral of fn).
State the dominated convergence theorem
Let (fn) be a sequence of measurable functions such that fn(x) converges a.e. to a limit f(x) and there is an integrable function g such that, for each n, |fn(x)| ≤ g(x) a.e. Then f is integrable, and the integral of f = lim(n→∞) of the integral of fn.
State the bounded convergence theorem
Let I be a bounded interval, (fn) be a sequence in L1(I) converging a.e. to f, and suppose that there is a constant c such that |fn(x)| ≤ c a.e., for all n. Then f ∈ L1 (I), and the integral over I of f = limn→∞ of the integral over I of fn.
State Beppo-Levi’s theorem
Let (gn) be a sequence of integrable functions such that the sum over n to ∞ of the integral of |gn| < ∞. Then the sum over n to ∞ of gn converges a.e. to an integrable function, and the integral of the sum of gn = the sum of the integral of gn.
State the continuous parameter DCT
Let I and J be intervals in R, and f: IxJ → R be a function such that:
1. For each y in J, x → f(x, y) is integrable over I.
2. For each y in J, lim(y’→y)f(x, y’) = f(x, y) a.e. in x
3. There exists an integrable function g on I such that for each y in J, |f(x, y)| ≤ g(x) a.e. in x.
Define F(y) = integral over I of f(x,y) dx for y ∈ J. Then F is continuous on J.
State the theorem of Tonelli
Let f: R^2 → [0, ∞] be measurable. Then:
1. x → f(x, y) is measurable for almost all y.
2. y → integral of f(x, y) dx over R (defined a.e.) is measurable.
3. The integral over R^2 of f(x, y) d(x, y) = the integral of (the integral of f(x, y) dx over R) dy over R.
State Fubini’s theorem
Let f: R^2 → R be integrable. Then, for almost all y, the function x → f(x, y) is integrable. Moreover, defining (for almost all y) by F(y) = the integral of f(x, y) dx, then F is integrable, and the integral over the integral of (the integral of f(x, y) dx) dy = R^2 of f(x, y)d(x, y) = the integral of (the integral of f(x, y) dy) dx.
State Tonelli’s Theorem
Let f: R^2 → R be a measurable function, and suppose that either of the following repeated integrals is finite: the integral of (the integral of |f(x, y)| dx) dy, integral of (the integral of |f(x, y)| dy) dx. Then f is integrable. Hence, Fubini’s Theorem is applicable to both f and |f|.
Give the Jacobian Theorem
Let E’ be an open subset of R^2, T: E’ → R^2 be a one-to-one differentiable function of E’ onto a subset E of R^2, and f: E → R be a function. Then f is integrable over E if and only if (f◦T)|det(J_T)| is integrable over E’. In that case, the integral over E of f = the integral over E’ of (f◦T)|det(J_T)|.
Define an L*p space
The space of functions f such that |f|^p is integrable.
Define equivalent functions on an L*p space
Given f,g ∈ L*p, let f ~ g if f = g a.e.
Define an Lp space
The quotient space of L*p/N, where N is the set of functions which are 0 a.e.
Define the norm on an Lp space
On Lp, || f||p = [the integral of |f^p|]^(1/p).
State Minkowski inequality
For p ≥ 1 and f, g ∈ Lp, ||f + g||p ≤ ||f||p + ||g||p.
Define the L*∞ space
The set of functions such that f is measurable and there exists C such that |f| ≤ C a.e.
Define the norm on L∞ space
The infimum of C such that |f| ≤ C.
State Holder’s inequality
Let p, q ∈ [1, ∞] with 1/p + 1/q = 1 (where we interpret 1/∞ = 0). Let f ∈ Lp and g ∈ Lq. Then fg ∈ L1 and ||fg||1 ≤ (||f||p)(||g||q).
State Egorov’s Theorem
Suppose that fn → f a.e. Let E be a measurable set with m(E) < ∞ and let ε > 0. Then there is a measurable subset F of E with m(E\F) < ε such that fn → f uniformly on F. In particular, ||fn – f||Lp(F) → 0 for all p ≥ 1.
Define the Fourier transformation
The Fourier transform of an integrable function f is given by the integral over R of f(x)e^(-isx)dx.
