theorems

absolute continuity of measures

Let \(ν\) be a finite signed measure and \(μ\) a positive measure on a measurable space \((X, 𝔐)\). Then \(ν ≪ μ\) if and only if for every \(ε>0\) there is a \(δ > 0\) such that \(|ν E|<ε\) whenever \(μ E < δ\). (Folland [Fol99] Lem 3.5.)

AC implies BV

If \(f ∈ AC[a,b]\) then \(f ∈ BV[a, b]\). (Royden [Roy88] Lem 11, Sec 5.4.)

AC implies a.e. differentiability

If \(f ∈ AC[a,b]\), then \(f'\) exists for almost every \(x∈ [a,b]\).

AC and a.e. zero derivative implies constant

If \(f ∈ AC[a,b]\) and \(f'(x) = 0\) a.e., then \(f\) is constant.

AC is equivalent to being an indefinite integral

\(F ∈ AC[a,b]\) iff \(F\) is an indefinite integral iff \(F(x) = ∫_a^x F'(t) \, dt + F(a)\). (Royden [Roy88] Thm 14, Sec 5.4.)

Banach space of bounded linear maps with complete codomain

If \(X\) and \(Y\) are normed linear spaces and \(Y\) is complete (i.e., a Banach space), then \(𝔅(X,Y)\) is complete (i.e., a Banach space).

Banach-Steinhaus theorem

Let \(X\) be a Banach space, and \(Y\) a normed linear space. Let \(ℱ\) be a family of bounded linear transformations each of which maps \(X\) to \(Y\).

If for each \(x ∈ X\) the set \(\{\|T x\|_Y ∣ T ∈ ℱ\}\) is bounded, then \(\{\|T\| : T ∈ ℱ\}\) is bounded.

This theorem is sometimes called the principle of uniform boundedness or the uniform boundedness principle.

Bolzano-Weierstrass theorem

See the Compactness theorem.

bounded convergence theorem

Let \(\{f_n\}\) be a sequence of measurable functions on a set of finite measure \(E\). Suppose \(\{f_n\}\) is uniformly pointwise bounded on \(E\); that is, \(∃ M > 0\) such that \(|f_n| < M\) on \(E\) for all \(n\). If \(f_n\) converges to \(f\) pointwise on \(E\), then

\[\lim\limits_{n→∞} ∫_E f_n = ∫_E \lim\limits_{n→∞} f_n = ∫_E f.\]

bounded linear transformations

If \(X\) and \(Y\) are normed linear spaces and \(T: X → Y\) is a linear transformation, the following are equivalent:

1. \(T\) is continuous.

2. \(T\) is continuous at \(0\).

3. \(T\) is bounded.

Caratheodory’s theorem

If \(μ^∗\) is an outer measure on \(X\), the collection \(𝔐\) of \(μ^∗\)-measurable sets is a (σ-algebra, and the restriction of \(μ^∗\) to \(𝔐\) is a complete measure.

closed graph theorem

If \(X\) and \(Y\) are Banach spaces and \(T: X → Y\) is a linear mapping, then the graph \(Γ(T) := \{(x,y) ∈ X × Y ∣ y = T x\}\) is closed if and only if \(T\) is bounded. (For proof see the solution to Problem 60.)

compactness theorem

A theorem that goes by various names (e.g., Bolzano-Weierstrass, Heine-Borel) is the following (cf. [Fol99] Thm. 0.25, [Con78] Cor. 4.9):

If \(E\) is a subset of a metric space \((X, d)\), then the following are equivalent.

1. \(E\) is complete and totally bounded.

2. Every infinite set in \(E\) has a limit point in \(E\);

3. (Bolzano-Weierstrass) Every sequence in \(E\) has a subsequence that converges to a point of \(E\).

4. (Heine-Borel) If \(\{V_\alpha\}\) is a cover of \(E\) by open sets, there is a finite set \(\{\alpha_1, \dots, \alpha_n\}\) such that \(\{V_{\alpha_i}\}_{i=1}^n\) covers \(E\).

continuity theorem

If \(f\) is continuous function in a compact set, then it is uniformly continuous in that set.

continuous and measurable functions

Let \(Y\) and \(Z\) be topological spaces, and let \(g: Y → Z\) be a continuous function.

1. If \(X\) is a topological space, if \(f: X → Y\) is continuous, and if \(h = g ∘ f\), then \(h: X → Z\) is continuous.

2. If \(X\) is a measurable space, if \(f: X → Y\) is a measurable function, and if \(h = g ∘ f\), then \(h: X → Z\) is measurable.

continuous functions on [-1,1] are dense in L₁[-1,1]

\(\overline{C[-1,1]} = L_1[-1,1]\).

continuous linear transformations

See bounded linear transformations.

derivative of the integral

Let \(f\) be an integrable function on \([a, b]\), and suppose that

\[F(x) = F(a) + ∫_a^x f(t) \, dt.\]

Then \(F'(x) = f(x)\) for almost all \(x\) in \([a,b]\). (Royden [Roy88] Thm 10, Sec 5.3.)

differentiability of increasing functions

If \(f: ℝ → ℝ\) is a monotone increasing function on the interval \([a,b]\), then \(f\) is differentiable a.e. on \([a,b]\), the derivative \(f'\) is measurable, and \(∫_a^b f'\, dm ≤ f(b) - f(a)\). (See Royden [Roy88] page 100-101.)

dominated convergence theorem

Let \(\{f_n\}\) be a sequence of measurable functions on \((X,𝔐,μ)\) such that \(f_n → f\) a.e. If there is another sequence of measurable functions \(\{g_n\}\) satisfying

1. \(g_n → g\) a.e.,

2. \(∫ g_n → ∫ g < ∞\), and

3. \(|f_n(x)| ≤ g_n(x) \quad (x ∈ X; n= 1, 2, \ldots)\),

then \(f ∈ L_1(X,𝔐,μ)\), \(∫ f_n → ∫ f\), and \(\|f_n - f\|_1 → 0\).

dominated convergence theorem (version 2)

Let \(\{f_n\}\) be a sequence of measurable functions on \((X,𝔐,μ)\) such that \(\lim_{n→∞}f_n(x) → f(x)\) exists for almost every \(x ∈ X\). If there exists \(g ∈ L_1(μ)\) such that for all \(n=1,2,\dots\) we have \(|f_n(x)| ≤ g(x)\) for almost every \(x ∈ X\), then

1. \(f ∈ L_1(μ)\),

2. \(∫_X|f_n - f| \, dμ → 0\), and

3. \(∫_X f_n \, dμ → ∫_X f \, dμ\).

Egoroff’s theorem

Suppose \((X, 𝔐, μ)\) is a measure space, \(E ∈ 𝔐\) is a set of finite measure, and \(\{f_n\}\) is a sequence of measurable functions such that \(f_n(x) → f(x)\) for almost every \(x ∈ E\). Then for all \(ε > 0\) there is a measurable set \(A ⊆ E\) such that \(f_n → f\) uniformly on \(A\) and \(μ (E - A) < ε\).

Fatou’s lemma

If \(f_n ≥ 0\) \((n = 1, 2, \dots)\) is a sequence of nonegative measurable functions, then \(∫ \lim \inf f_n ≤ \lim \inf ∫ f_n\).

Fubini’s theorem

Assume \((X, 𝔐, μ)\) and \((Y, 𝔑, ν)\) are σ-finite measure spaces, and \(f(x,y)\) is a \((𝔐 ⊗ 𝔑)\)-measurable function on \(X × Y\).

1. If \(f(x,y) ≥ 0\), and if \(φ(x)= ∫_Y f(x,y) \, dν(y)\) and \(ψ(y)= ∫_X f(x,y) \, dμ(x)\), then \(φ\) is \(𝔐\)-measurable, \(ψ\) is \(𝔑\)-measurable, and

   (62)\[∫_X φ \, dμ = ∫_{X × Y} f(x,y) \, d(μ × ν) = ∫_Y ψ \, dν.\]

2. If \(f: X × Y → ℂ\) and if one of

   \[∫_Y ∫_X |f(x,y)| \, dμ(x)\, dν(y) < ∞ \quad \text{ or } \quad ∫_X ∫_Y |f(x,y)| \, dν(y)\, dμ(X) < ∞\]

   holds, then so does the other, and \(f ∈ L_1(μ × ν)\).

3. If \(f ∈ L_1(μ × ν)\), then,

   1. for almost every \(x ∈ X,\, f(x, y) ∈ L_1(ν)\),

   2. for almost every \(y ∈ Y,\, f(x, y) ∈ L_1(μ)\),

   3. \(φ(x)= ∫_Y f \, dν\) is defined almost everywhere (by 1.), moreover \(φ ∈ L_1(μ)\),

   4. \(ψ(y)= ∫_X f \, dμ\) is defined almost everywhere (by 2.), moreover \(ψ ∈ L_1(ν)\), and

   5. equation (62) holds.

1

Fundamental theorem of calculus

If \(-∞ < a < b < ∞\) and \(f: [a,b] → ℂ\), then the following are equivalent.

1. \(f ∈ AC[a,b]\)

2. \(f(x) - f(a) = ∫_a^x g(t) \, dt\) for some \(g∈ L_1([a,b], m)\).

3. \(f\) is differentiable a.e. on \([a,b]\), \(f' ∈ L_1([a,b],m)\) and \(∫_a^x f' \, dm = f(x) - f(a)\).

Hahn-Banach theorem

Suppose \(X\) is a normed linear space, \(Y ≤ X\) is a subspace, and \(T:Y → ℝ\) is a bounded linear functional.

Then there exists a bounded linear functional \(T̄: X → ℝ\) such that \(T̄ y = T y\) for all \(y ∈ Y\), and such that \(\| T̄ \|_X = \| T \|_Y\), where \(\| T̄ \|_X\) and \(\| T \|_Y\) are the usual operator norms,

\[\|T̄\|_X = \sup\{|T̄ x|: x ∈ X, \|x\| ≤ 1\} \quad \text{ and } \quad \|T\|_Y = \sup\{|T x|: x ∈ Y, \|x\| ≤ 1\}.\]

Heine-Borel theorem

See the Compactness theorem.

Hellinger-Toeplitz theorem

Every everywhere-defined self-adjoint linear operator on a Hilbert space is bounded.

Equivalently, if \(T\) is an everywhere-defined linear transformation on a Hilbert space \(ℋ\), and if \(T\) is self-adjoint (i.e., \(∀ x, y ∈ ℋ, ⟨x, Ty⟩ = ⟨Tx, y⟩\)), then \(T\) is bounded.

Hölder’s inequality

Let \(p\) and \(q\) be conjugate exponents and \((X, 𝔐, μ)\) a measure space.

Assume \(f, g ≥ 0\) are nonnegative, measurable functions on \(X\) with range in \([0,∞]\).

1. If \(1 < p < ∞\), then

   (63)\[∫_X fg \, dμ ≤ \left( ∫_X f^p \, dμ\right)^{1/p} \left(∫_X g^q \, dμ\right)^{1/q}, \text{ and}\]

2. If \(p = ∞\), \(f ∈ L_∞\), and \(g ∈ L_1\), then \(|f g| ≤ \|f\|_∞ |g|\), so \(\|f g\|_1 ≤ \|f\|_∞ \|g\|_1\).

We stated the theorem for nonnegative extended real-valued functions, but if \(f, g: X → [-∞, ∞]\), then we apply the theorem to \(|f|\) and \(|g|\).

(See also Minkowski’s inequality.)

integrability of a measurable function

Let \(f: X → [-∞, ∞]\) be a measurable function on \(X\). Then the positive and negative parts of \(f\) (denoted by \(f^+\) and \(f^-\), respectively) are integrable over \(X\) if and only if \(|f|\) is integrable over \(X\).

Proof.

Assume \(f^+\) and \(f^-\) are integrable (nonnegative) functions. By the linearity of integration for nonnegative functions, \(|f| = f^+ + f^-\) is integrable. Conversely, if \(|f|\) is integrable, then since \(0 < f^+ < |f|\) and \(0 < f^- < |f|\), we infer from the monotonicity of integration for nonnegative functions that both \(f^+\) and \(f^-\) are integrable.

integral extensionality

Suppose \(f\) and \(g\) are integrable functions such that for all measurable \(E\) we have \(∫_E f = ∫_E g\). Then \(f = g\) \(μ\)-a.e.

inverse function theorem

Let \(f: E → ℝ^n\) be a \(C^1\)-mapping of an open set \(E ⊆ ℝ^n\). Suppose that \(f'(a)\) is invertible for some \(a ∈ E\) and that \(f(a)=b\). Then,

1. there exist open sets \(U\) and \(V\) in \(ℝ^n\) such that \(a ∈ U\), \(b ∈ V\), and \(f\) maps \(U\) bijectively onto \(V\), and

2. if \(g\) is the inverse of \(f\) (which exists by (i)), defined on \(V\) by \(g(f(x))=x\), for \(x ∈ U\), then \(g ∈ C^1(V)\).

See also Rudin, Principles of Mathematical Analysis [Rud76].

inverse mapping theorem

Let \(X, Y\) be Banach spaces. A continuous bijection \(T: X → Y\) has a continuous inverse.

That is, if \(G\) is an open subset of \(X\), then \((T^{-1})^{-1}(G)\) is an open subset of \(Y\).

Lusin’s theorem

Fix \(ε > 0\). If \(f\) is a measurable function that vanishes off a set of finite measure then there exists \(g ∈ C_c(X)\) such that \(μ \{x ∣ f(x) ≠ g(x)\} < ε\). Moreover, we may arrange it so that \(\|g\|_{\sup} ≤ \|f\|_{\sup}\).

mean value theorem

If a real-valued function \(f\) is continuous on the (closed, bounded) interval \([a, b]\) and differentiable on \((a, b)\), then there exists \(c ∈ (a,b)\) such that

\[f'(c) = \frac{f(b) - f(a)}{b-a}.\]

mean value theorem (version 2)

If a real-valued function \(f\) is continuous on the \(closed <closed set>\) bounded interval \([c, d]\) and differentiable on its interior \((c, d)\) with \(f' > α\) on \((c, d)\), then \(α ⋅ (d - c) < f(d)-f(c)\).

measurability of upper limit

If \(f_n: X → [-∞,∞]\) is measurable for each \(n∈ ℕ\), \(g = \sup\limits_{n ≥ 0} f_n\), and \(h = \limsup\limits_{n→ ∞} f_n\), then \(g\) and \(h\) are measurable.

Proof.

\(g^{-1}((α, ∞]) = ⋃_{n∈ ℕ} f^{-1}((α, ∞])\), so \(g\) is measurable. The same result holds with inf in place of sup, and since

\[h = \inf\limits_{k≥ 0} \left\{ \sup\limits_{i≥ k} f_i\right\},\]

it follows that \(h\) is measurable.

Minkowski’s inequality

Let \(1 < p < ∞\) and \(q\) be conjugate exponents and let \((X, 𝔐, μ)\) be a measure space. If \(f, g\) are measurable functions on \(X\), then

\[\left(∫_X (|f|+|g|)^p \, dμ\right)^{1/p} ≤ \left( ∫_X |f|^p \, dμ\right)^{1/p} + \left(∫_X |g|^p \, dμ\right)^{1/p}.\]

(See also Hölder’s inequality.)

monotone convergence theorem

Let \(\{f_n\}\) be a sequence of measurable functions on \(X\), and suppose that, for every \(x ∈ X\),

1. \(0 ≤ f_1(x) ≤ f_2(x) ≤ \cdots ≤ ∞\),

2. \(f_n(x) → f(x)\) as \(n → ∞\).

Then \(f\) is measurable, and \(∫_X f_n \, dμ → ∫_X f \, dμ\) as \(n → ∞\).

open mapping theorem

A surjective bounded linear transformation from one Banach space onto another is an open mapping.

properties of the Lebesgue integral

Assume \(f\) and \(g\) are measurable functions defined on the set \(X\). Assume also that \(A ⊆ B ⊆ X\), \(E⊆ X\), and \(A\), \(B\), \(E\) are measurable sets. Finally, let \(0≤ c < ∞\) be an arbitrary positive constant. Then the Lebesgue integral has the following properties:

1. If \(0 ≤ f ≤ g\), then \(∫_E f ≤ ∫_E g\).

2. If \(f≥ 0\), then \(∫_A f ≤ ∫_B f\).

3. If \(f≥ 0\), then \(∫_E cf = c∫_E f\).

4. If \(f(x) = 0\) for every \(x ∈ E\), then \(∫_E f = 0\), even if \(E\) has infinite measure.

5. If \(E\) is negligible, then \(∫_E f = 0\), even if \(f(x) = ∞\) for all \(x ∈ E\).

6. If \(f≥ 0\), then \(∫_E f = ∫_X χ_E f\).

properties of measures

Let \((X, 𝔐, μ)\) be a measure space. Then

1. \(μ(A_1 ∪ \cdots ∪ A_n) = μ A_1 + \cdots + μ A_n\) if \(A_1, \dots, A_n\) are pairwise disjoint sets in \(𝔐\);

2. \(μ ∅ = 0\);

3. \(μ A ≤ μ B\) if \(A ⊆ B\) and \(A, B ∈ 𝔐\);

4. \(μ A_n → μ A\) as \(n→ ∞\) if \(∀ n, A_n ∈ 𝔐\) and \(A_1 ⊆ A_2 ⊆ \cdots\) and \(A = ⋃_n A_n\).

5. \(μ A_n → μ A\) as \(n→ ∞\) if \(∀ n, A_n ∈ 𝔐\) and \(A_1 ⊇ A_2 ⊇ \cdots\) and \(A = ⋂_n A_n\) and \(μ A_1 < ∞\).

Radon-Nikodym theorem

If \(λ\) and \(m\) are σ-finite positive measures on a σ-algebra \(Σ\) and if \(λ ≪ m\), then there is a unique \(g ∈ L_1(dm)\) such that \(∀ E ∈ Σ\), \(λ E = ∫_E g \, dm\).

Radon-Nikodym theorem (full version)

Let \((X, 𝔐, μ)\) be a measure space and assume \(μ\) is a positive σ-finite measure.

If \(λ\) is a complex measure on \(𝔐\), then

1. there is then a unique pair of complex measures \(λ_a\) and \(λ_s\) on \(𝔐\) such that

   \[λ = λ_a + λ_s, \quad λ_a ≪ μ, \quad λ_s ⟂ μ;\]

   if \(λ\) happens to be a real positive finite measure, then so are \(\lambda_a\) and \(\lambda_s\);

2. there is a unique \(h ∈ L_1(μ)\) such that

   \[λ_a E = ∫_E h \, dμ \quad ∀ E ∈ 𝔐.\]

The pair \((λ_a, λ_s)\) is called the Lebesgue decomposition of \(λ\) relative to \(μ\).

Radon-Nikodym corollary

Suppose \(ν\) is a σ-finite complex measure and \(μ, λ\) are \(σ\)-finite measures on \((X, 𝔐)\) such that \(ν ≪ μ ≪ λ\). Then

1. If \(g ∈ L_1(ν)\), then \(g \frac{dν}{dμ} ∈ L_1(μ)\) and

   \[∫ g\, dν = ∫ g \frac{dν}{dμ} \, dμ.\]

2. \(ν ≪ λ\), and

   \[\frac{dν}{dλ} = \frac{dν}{dμ} \frac{dμ}{dλ} \quad λ\text{-a.e.}\]

Riesz representation theorem

Let \(X\) be a locally compact Hausdorff space, and let \(Λ\) be a positive linear functional on \(C_c(X)\).

There exists a σ-algebra \(𝔐\) in \(X\) that contains all Borel sets in \(X\), and there exists a unique positive measure \(μ\) on \(𝔐\) that represents \(Λ\) in the following sense:

1. \(Λ f = ∫_X f \, dμ\) for every \(f ∈ C_c(X)\), and the following additional properties hold:

2. \(μ K < ∞\) for every compact set \(K ⊆ X\);

3. for every \(E ∈ 𝔐\), we have

   \[μ E = \inf \{ μ V ∣ E ⊆ V, V \text{ open}\};\]

4. the relation

   \[μ E = ∑ \{ μ K ∣ K ⊆ E, K \text{ compact}\}\]

   holds for every open set \(E\), and for every \(E ∈ 𝔐\) with \(μ E < ∞\);

5. if \(E ∈ 𝔐\), \(A ⊆ E\), and \(μ E = 0\), the \(A ∈ 𝔐\).

(See also Rudin, Real and Complex Analysis [Rud87] 2.14.)

Riesz representation theorem (version 2)

Suppose \(1 < p < ∞\) and \(\frac{1}{p} + \frac{1}{q} = 1\). If \(Λ\) is a linear functional on \(L_p\), then there is a unique \(g ∈ L_q\) such that \(Λ f = ∫ f g \, dμ\) for all \(f ∈ L_p\).

Stone-Weierstrass theorem

Let \(X\) be a compact Hausdorff space and let \(𝔄\) be a subalgebra of \(C(X,ℝ)\) that separates the points of \(X\) and contains the constant functions. Then \(𝔄\) is dense in \(C(X, ℝ)\).

Stone-Weierstrass theorem (version 2)

Let \(X\) be a compact Hausdorff space and let \(𝒜\) be a closed subalgebra of functions in \(C(X,ℝ)\) that separates the points of \(X\). Then either \(𝒜 = C(X,ℝ)\), or \(𝒜 = \{f ∈ C(X,ℝ) ∣ f(x_0) = 0\}\) for some \(x_0 ∈ X\). The first case occurs iff \(𝒜\) contains the constant functions.

Tonelli’s theorem

Let \((X, 𝔄, μ)\) and \((Y, 𝔅, ν)\) be σ-finite complete measure spaces and let \(f\) be a nonnegative \((μ × ν)\)-measurable function on \(X × Y\). Then,

1. for a.e. \(x ∈ X\), the function \(y ↦ f(x,y)\) is \(ν\)-measurable, and the function defined a.e. on \(X\) by \(φ(x) := ∫ f (x, y) \, dν(y)\) is \(μ\)-measurable;

2. for a.e. \(y ∈ Y\), the function \(x ↦ f(x,y)\)) is \(μ\)-measurable and the function defined a.e. on \(Y\) by \(ψ(y) := ∫ f (x, y) \, dμ(x)\) is \(ν\)-measurable;

3. If \(∫_X \bigl[∫_Y f (x, y) \, dν(y) \bigr] dμ(x) < ∞\), then \(f\) is integrable over \(X × Y\) with respect to \(μ × v\) and

   \[∫_Y \bigl[∫_X f (x, y) \, dμ(x) \bigr] dν(y) = ∫_{X × Y} f \, d(μ × ν) = \int_X \bigl[∫_Y f (x, y) \, dν(y) \bigr] dμ(x).\]

Notice that Tonelli’s theorem concerns nonnegative functions. This makes it less generally applicable (but sometimes easier to apply) than Fubini’s theorem. 1

Tychonoff’s theorem

Let \({X_α}_{α∈𝔄}\) be a be a collection of compact topological spaces indexed by a set \(𝔄\). Then the Cartesian product \(∏_{α∈𝔄}X_α\) with the product topology also is compact.

uniform boundedness principle

See the Banach-Steinhaus theorem.

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Footnotes

1(1,2)

The most useful among the many versions of the theorem bearing the name Fubini and/or Tonelli is the one that appears in Rudin’s Real and Complex Analysis [Rud87]. Rudin begins by assuming only that the function \(f(x,y)\) is measurable with respect to the product σ-algebra \(𝔐 ⊗ 𝔑\). Then, in a single, combined Fubini-Tonelli theorem, you get everything you need to answer all standard questions about integration with respect to product measure.

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Complex Analysis Exams

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