Integration

Question 1

Definie addition by infinity

Answer 1

For all real x, x + ∞ = ∞ + x = ∞

Question 2

Define subtraction by infinity

Answer 2

For all real x, x − ∞ = −∞ + x = −∞,

Question 3

Define multiplication by infinity

Answer 3

For all real x, x.∞ = ∞.x = (−x).(−∞) and x.∞ = ∞ for x>0, x.∞ = -∞ for x<0 and x.∞ = 0 for x=0.

Question 4

Define the limsup of a sequence

Answer 4

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.

Question 5

Define the outer measure

Answer 5

m*(A) = inf(The sum of |I_n|, where I_n are intervals whose union contains A).

Question 6

Define a null set

Answer 6

A set A is null if m*(A) = 0.

Question 7

Define the Cantor set

Answer 7

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.

Question 8

Define almost everywhere

Answer 8

A property hold almost everywhere if the set A not satisfying the property is null.

Question 9

Define Lebesgue measurable

Answer 9

A subset E of R is Lebesgue measurable if m(A) = m(A∩E) + m*(A\E).

Question 10

Define a σ-algebra

Answer 10

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.

Question 11

Define a measure

Answer 11

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.

Question 12

Define a measure space

Answer 12

A triple (Ω, F, µ) is a measure space.

Question 13

Define a finite measure

Answer 13

A measure µ is finite if µ(Ω) < ∞.

Question 14

Define a probability measure

Answer 14

A measure µ is a probability measure if µ(Ω) = 1.

Question 15

Define the Borel measure

Answer 15

The smallest σ-algebra containing all the intervals.

Question 16

Define a measurable function

Answer 16

f is F-measurable if and only if f^(−1)G ∈ F for all G in the measure space.

Question 17

Define a simple function

Answer 17

A function which takes a finite number of values.

Question 18

Define the standard form of a simple function

Answer 18

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.

Question 19

Define the integral of a non-negative simple function

Answer 19

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).

Question 20

Define the integral of a non-negative function

Answer 20

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}.

Question 21

Define integrable

Answer 21

A function is integrable over a set if the integral over that set is finite.

Question 22

State the monotone convergence theorem

Answer 22

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.

Question 23

State the baby monotone convergence theorem

Answer 23

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.

Question 24

State the comparison test

Answer 24

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.

Question 25

State the fundamental theorem of calculus

Answer 25

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).

Question 26

State integration by parts

Answer 26

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).

Question 27

State integration by substitution

Answer 27

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.

Question 28

State Fatou's lemma

Answer 28

Let (fn) be a sequence of non-negative measurable functions. Then the integral of liminf(fn) ≤ liminf(the integral of fn).

Question 29

State the dominated convergence theorem

Answer 29

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.

Question 30

State the bounded convergence theorem

Answer 30

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.

Question 31

State Beppo-Levi's theorem

Answer 31

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.

Question 32

State the continuous parameter DCT

Answer 32

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.

Question 33

State the theorem of Tonelli

Answer 33

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.

Question 34

State Fubini's theorem

Answer 34

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.

Question 35

State Tonelli's Theorem

Answer 35

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|.

Question 36

Give the Jacobian Theorem

Answer 36

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)|.

Question 37

Define an L*p space

Answer 37

The space of functions f such that |f|^p is integrable.

Question 38

Define equivalent functions on an L*p space

Answer 38

Given f,g ∈ L*p, let f ~ g if f = g a.e.

Question 39

Define an Lp space

Answer 39

The quotient space of L*p/N, where N is the set of functions which are 0 a.e.

Question 40

Define the norm on an Lp space

Answer 40

On Lp, || f||p = [the integral of |f^p|]^(1/p).

Question 41

State Minkowski inequality

Answer 41

For p ≥ 1 and f, g ∈ Lp, ||f + g||p ≤ ||f||p + ||g||p.

Question 42

Define the L*∞ space

Answer 42

The set of functions such that f is measurable and there exists C such that |f| ≤ C a.e.

Question 43

Define the norm on L∞ space

Answer 43

The infimum of C such that |f| ≤ C.

Question 44

State Holder's inequality

Answer 44

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).

Question 45

State Egorov's Theorem

Answer 45

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.

Question 46

Define the Fourier transformation

Answer 46

The Fourier transform of an integrable function f is given by the integral over R of f(x)e^(-isx)dx.