Integration

Measurable Partition

Suppose that

𝒮

is a

σ

-algebra on a set

X

. An $𝒮$ -partition of

X

is a finite collection

A_{1}, \dots, A_{m}

of pairwise disjoint sets in

𝒮

such that

A_{1} \cup \dots \cup A_{m} = X .

Lower Lebesgue Sum

Suppose that

(X, 𝒮, μ)

is a measure space,

f : X \to [0, \infty]

𝒮

-measurable, and

P = (A_{1}, \dots, A_{m})

is an

𝒮

-partition of

X

. The lower Lebesgue sum of

f

over

P

L (f, P) : = \sum_{j = 1}^{m} μ (A_{j}) \inf_{A_{j}} f .

Integral of a Nonnegative Measurable Function

Suppose that

(X, 𝒮, μ)

is a measure space and

f : X \to [0, \infty]

𝒮

-measurable. The integral of

f

with respect to

μ

\int f d μ : = \sup {L (f, P) : P is an 𝒮 -partition of X} .

Integral of a Characteristic Function

Suppose that

(X, 𝒮, μ)

is a measure space and

E \in 𝒮

. Then

\int χ_{E} d μ = μ (E) .

Integral of a Simple Function

Suppose that

(X, 𝒮, μ)

is a measure space,

E_{1}, \dots, E_{n} \in 𝒮

are pairwise disjoint, and

c_{1}, \dots, c_{n} \in [0, \infty]

. Then

\int (\sum_{k = 1}^{n} c_{k} χ_{E_{k}}) d μ = \sum_{k = 1}^{n} c_{k} μ (E_{k}) .

Integration of Nonnegative Functions is Order Preserving

Suppose that

(X, 𝒮, μ)

is a measure space and

f, g : X \to [0, \infty]

are

𝒮

-measurable. If

f (x) \leq g (x)

for every

x \in X

, then

\int f d μ \leq \int g d μ .

Monotone Convergence Theorem

Suppose that

(X, 𝒮, μ)

is a measure space and

0 \leq f_{1} \leq f_{2} \leq \dots

is an increasing sequence of

𝒮

-measurable functions. Define

f : X \to [0, \infty]

f (x) : = \lim_{k \to \infty} f_{k} (x) .

Then

\lim_{k \to \infty} \int f_{k} d μ = \int f d μ .

Additivity of Integration for Nonnegative Functions

Suppose that

(X, 𝒮, μ)

is a measure space and

f, g : X \to [0, \infty]

are

𝒮

-measurable. Then

\int (f + g) d μ = \int f d μ + \int g d μ .

Positive Part of a Function

Suppose that

f : X \to [- \infty, \infty]

. The positive part of

f

is the function

f^{+} : X \to [0, \infty]

defined by

f^{+} (x) : = {\begin{matrix} f (x) & a m p; f (x) \geq 0, \\ 0 & a m p; f (x) < 0. \end{matrix}

Negative Part of a Function

Suppose that

f : X \to [- \infty, \infty]

. The negative part of

f

is the function

f^{-} : X \to [0, \infty]

defined by

f^{-} (x) : = {\begin{matrix} 0 & a m p; f (x) \geq 0, \\ - f (x) & a m p; f (x) < 0. \end{matrix}

Integral of an Extended Real-Valued Function

Suppose that

(X, 𝒮, μ)

is a measure space and

f : X \to [- \infty, \infty]

𝒮

-measurable. If at least one of

\int f^{+} d μ

and

\int f^{-} d μ

is finite, define

\int f d μ : = \int f^{+} d μ - \int f^{-} d μ .

Additivity of Integration

Suppose that

(X, 𝒮, μ)

is a measure space and

f, g : X \to ℝ

are

𝒮

-measurable. If

\int | f | d μ < \infty and \int | g | d μ < \infty,

then

\int (f + g) d μ = \int f d μ + \int g d μ .

Integration on a Subset

Suppose that

(X, 𝒮, μ)

is a measure space,

E \in 𝒮

, and

f : X \to [- \infty, \infty]

𝒮

-measurable. Define

\int_{E} f d μ : = \int χ_{E} f d μ

whenever the integral on the right is defined.

Almost Every

Suppose that

(X, 𝒮, μ)

is a measure space. A set

E \in 𝒮

contains $μ$ -almost every element of

X

μ (X ∖ E) = 0.

Bounded Convergence Theorem

Suppose that

(X, 𝒮, μ)

is a measure space with

μ (X) < \infty

. Suppose that

f_{1}, f_{2}, \dots : X \to ℝ

are

𝒮

-measurable and converge pointwise to

f : X \to ℝ

. If there exists

c \in ℝ^{+}

such that

| f_{k} (x) | \leq c

for every

k \in ℤ^{+}

and every

x \in X

, then

\lim_{k \to \infty} \int f_{k} d μ = \int f d μ .

Dominated Convergence Theorem

Suppose that

(X, 𝒮, μ)

is a measure space,

f : X \to [- \infty, \infty]

𝒮

-measurable, and

f_{1}, f_{2}, \dots : X \to [- \infty, \infty]

are

𝒮

-measurable. Suppose that

f_{k} \to f

for almost every

x \in X

. If there exists an

𝒮

-measurable

g : X \to [0, \infty]

such that

\int g d μ < \infty and | f_{k} (x) | \leq g (x)

for every

k \in ℤ^{+}

and almost every

x \in X

, then

\lim_{k \to \infty} \int f_{k} d μ = \int f d μ .

Riemann Integrable iff Continuous Almost Everywhere

Suppose that

a < b

and

f : [a, b] \to ℝ

is bounded. Then

f

is Riemann integrable if and only if

| {x \in [a, b] : f is not continuous at x} | = 0.

Moreover, if

f

is Riemann integrable and

λ

is Lebesgue measure on

ℝ

, then

f

is Lebesgue measurable and

\int_{a}^{b} f = \int_{[a, b]} f d λ .

Lebesgue Integral on an Interval

Suppose that

- \infty \leq a < b \leq \infty

and

f : (a, b) \to ℝ

is Lebesgue measurable. Let

λ

be Lebesgue measure on

ℝ

. Define

\int_{a}^{b} f : = \int_{(a, b)} f d λ

and define

\int_{b}^{a} f : = - \int_{a}^{b} f

L^{1}

-Norm

Suppose that

(X, 𝒮, μ)

is a measure space and

f : X \to [- \infty, \infty]

𝒮

-measurable. Define

{‖ f ‖}_{1} : = \int | f | d μ .

L^{1}

-Space

Suppose that

(X, 𝒮, μ)

is a measure space. The Lebesgue space

L^{1} (μ)

L^{1} (μ) : = {f : X \to ℝ : f is 𝒮 -measurable and {‖ f ‖}_{1} < \infty} .

Approximation in

L^{1}

by Simple Functions

Suppose that

μ

is a measure and

f \in L^{1} (μ)

. For every

ϵ \in ℝ^{+}

, there exists a simple function

g \in L^{1} (μ)

such that

{‖ f - g ‖}_{1} < ϵ .

Step Function

A step function is a function

g : ℝ \to ℝ

of the form

g = a_{1} χ_{I_{1}} + \dots + a_{n} χ_{I_{n}},

where

I_{1}, \dots, I_{n}

are intervals of

ℝ

and

a_{1}, \dots, a_{n} \in ℝ ∖ {0}

Approximation in

L^{1}

by Continuous Functions

Suppose that

f \in L^{1} (ℝ)

. For every

ϵ \in ℝ^{+}

, there exists a continuous function

g : ℝ \to ℝ

such that

{‖ f - g ‖}_{1} < ϵ

and

{x \in ℝ : g (x) \neq 0}

is bounded.