.. _appendix-theorems:

===================
theorems
===================

.. contents:: :local:
    :depth: 1

.. glossary::

    absolute continuity of measures
      Let :math:`ν` be a finite signed measure and :math:`μ` a positive measure on a :term:`measurable space` :math:`(X, 𝔐)`. Then :math:`ν ≪ μ` if and only if for every :math:`ε>0` there is a :math:`δ > 0` such that :math:`|ν E|<ε` whenever :math:`μ E < δ`. (Folland :cite:`Folland:1999` Lem 3.5.)

    AC implies BV
      If :math:`f ∈ AC[a,b]` then :math:`f ∈ BV[a, b]`.  (Royden :cite:`Royden:1988` Lem 11, Sec 5.4.)

    AC implies a.e. differentiability
      If :math:`f ∈ AC[a,b]`, then :math:`f'` exists for almost every :math:`x∈ [a,b]`.

    AC and a.e. zero derivative implies constant
      If :math:`f ∈ AC[a,b]` and :math:`f'(x) = 0` a.e., then :math:`f` is constant.

    AC is equivalent to being an indefinite integral
      :math:`F ∈ AC[a,b]` iff :math:`F` is an indefinite integral iff :math:`F(x) = ∫_a^x F'(t) \, dt + F(a)`. (Royden :cite:`Royden:1988` Thm 14, Sec 5.4.)

    Banach space of bounded linear maps with complete codomain
      If :math:`X` and :math:`Y` are :term:`normed linear spaces <normed linear space>` and :math:`Y` is :term:`complete <complete set>` (i.e., a :term:`Banach space`), then :math:`𝔅(X,Y)` is complete (i.e., a :term:`Banach space`).

    Banach-Steinhaus theorem
      Let :math:`X` be a :term:`Banach space`, and :math:`Y` a :term:`normed linear space`. Let :math:`ℱ` be a family of :term:`bounded linear transformations <bounded linear transformation>` each of which maps :math:`X` to :math:`Y`.
    
      If for each :math:`x ∈ X` the set :math:`\{\|T x\|_Y ∣ T ∈ ℱ\}` is bounded, then :math:`\{\|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 :term:`Compactness theorem`.

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

      .. math:: \lim\limits_{n→∞} ∫_E f_n = ∫_E \lim\limits_{n→∞} f_n = ∫_E f.

    bounded linear transformations
      If :math:`X` and :math:`Y` are :term:`normed linear spaces <normed linear space>` and :math:`T: X → Y` is a :term:`linear transformation`, the following are equivalent:
      
      #. :math:`T` is :term:`continuous <continuous function>`.
      #. :math:`T` is :term:`continuous <continuous function>` at :math:`0`.
      #. :math:`T` is :term:`bounded <bounded linear transformation>`.
      
    Caratheodory's theorem
      If :math:`μ^∗` is an :term:`outer measure` on :math:`X`, the collection :math:`𝔐` of :math:`μ^∗`-:term:`measurable sets <measurable set>` is a (:term:`σ-algebra`, and the restriction of :math:`μ^∗` to :math:`𝔐` is a :term:`complete measure`.

    closed graph theorem
      If :math:`X` and :math:`Y` are :term:`Banach spaces <Banach space>` and :math:`T: X → Y` is a linear mapping, then the graph :math:`Γ(T) := \{(x,y) ∈ X × Y ∣ y = T x\}` is closed if and only if :math:`T` is bounded. (For proof see the solution to :numref:`Problem {number} <2004Apr_p4>`.)

    compactness theorem
      A theorem that goes by various names (e.g., Bolzano-Weierstrass, Heine-Borel) is the following (cf. :cite:`Folland:1999` Thm. 0.25, :cite:`Conway:1978` Cor. 4.9):
      
      If :math:`E` is a subset of a :term:`metric space` :math:`(X, d)`, then the following are equivalent.
  
      #. :math:`E` is :term:`complete <complete set>` and :term:`totally bounded`.
      #. Every infinite set in :math:`E` has a :term:`limit point` in :math:`E`;
      #. (Bolzano-Weierstrass) Every sequence in :math:`E` has a subsequence that converges to a point of :math:`E`.
      #. (Heine-Borel) If :math:`\{V_\alpha\}` is a :term:`cover` of :math:`E` by open sets, there is a finite set :math:`\{\alpha_1, \dots, \alpha_n\}` such that :math:`\{V_{\alpha_i}\}_{i=1}^n` covers :math:`E`.

    continuity theorem
      If :math:`f` is :term:`continuous function` in a :term:`compact set`, then it is :term:`uniformly continuous` in that set.

    continuous and measurable functions
      Let :math:`Y` and :math:`Z` be :term:`topological spaces <topological space>`, and let :math:`g: Y → Z` be a :term:`continuous function`.

      #. If :math:`X` is a topological space, if :math:`f: X → Y` is continuous, and if :math:`h = g ∘ f`, then :math:`h: X → Z` is continuous.

      #. If :math:`X` is a :term:`measurable space`, if :math:`f: X → Y` is a :term:`measurable function`, and if :math:`h = g ∘ f`, then :math:`h: X → Z` is measurable.

    continuous functions on [-1,1] are dense in L₁[-1,1]
      :math:`\overline{C[-1,1]} = L_1[-1,1]`.

    continuous linear transformations
      See :term:`bounded linear transformations`.

    derivative of the integral
      Let :math:`f` be an integrable function on :math:`[a, b]`, and suppose that

      .. math:: F(x) = F(a) + ∫_a^x f(t) \, dt.

      Then :math:`F'(x) = f(x)` for almost all :math:`x` in :math:`[a,b]`. (Royden :cite:`Royden:1988` Thm 10, Sec 5.3.)

    differentiability of increasing functions
      If :math:`f: ℝ → ℝ` is a :term:`monotone increasing function` on the interval :math:`[a,b]`, then :math:`f` is differentiable a.e. on :math:`[a,b]`, the derivative :math:`f'` is measurable, and :math:`∫_a^b f'\, dm ≤ f(b) - f(a)`. (See Royden :cite:`Royden:1988` page 100-101.)

    dominated convergence theorem
      Let :math:`\{f_n\}` be a sequence of :term:`measurable functions <measurable function>` on :math:`(X,𝔐,μ)` such that :math:`f_n → f` a.e. If there is another sequence of measurable functions :math:`\{g_n\}` satisfying

      #. :math:`g_n → g` a.e.,
      #. :math:`∫ g_n → ∫ g < ∞`, and
      #. :math:`|f_n(x)| ≤ g_n(x) \quad (x ∈ X; n= 1, 2, \ldots)`,

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

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

      #. :math:`f ∈ L_1(μ)`,
      #. :math:`∫_X|f_n - f| \, dμ → 0`, and
      #. :math:`∫_X f_n \, dμ → ∫_X f \, dμ`.

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

    Fatou's lemma
      If :math:`f_n ≥ 0` :math:`(n = 1, 2, \dots)` is a sequence of nonegative :term:`measurable functions <measurable function>`, then :math:`∫ \lim \inf f_n ≤ \lim \inf ∫ f_n`.

    Fubini's theorem
      Assume :math:`(X, 𝔐, μ)` and :math:`(Y, 𝔑, ν)` are :term:`σ-finite <σ-finite measure>` :term:`measure spaces <measure space>`, and :math:`f(x,y)` is a :math:`(𝔐 ⊗ 𝔑)`-:term:`measurable function` on :math:`X × Y`.

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

         .. math:: ∫_X φ \, dμ = ∫_{X × Y} f(x,y) \, d(μ × ν) = ∫_Y ψ \, dν.
           :label: fubini

      #. If :math:`f: X × Y → ℂ` and if one of

         .. math:: ∫_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 :math:`f ∈ L_1(μ × ν)`.

      #. If :math:`f ∈ L_1(μ × ν)`, then,

         #. for almost every :math:`x ∈ X,\, f(x, y) ∈ L_1(ν)`,
  
         #. for almost every :math:`y ∈ Y,\, f(x, y) ∈ L_1(μ)`,

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

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

         #. equation :eq:`fubini` holds.

      [1]_

    Fundamental theorem of calculus
      If :math:`-∞ < a < b < ∞` and :math:`f: [a,b] → ℂ`, then the following are equivalent.

      #. :math:`f ∈ AC[a,b]`
      #. :math:`f(x) - f(a) = ∫_a^x g(t) \, dt` for some :math:`g∈ L_1([a,b], m)`.
      #. :math:`f` is differentiable a.e. on :math:`[a,b]`, :math:`f' ∈ L_1([a,b],m)` and :math:`∫_a^x f' \, dm = f(x) - f(a)`.

    Hahn-Banach theorem
      Suppose :math:`X` is a :term:`normed linear space`, :math:`Y ≤ X` is a subspace, and :math:`T:Y → ℝ` is a :term:`bounded linear functional`.
      
      Then there exists a bounded linear functional :math:`T̄: X → ℝ` such that :math:`T̄ y = T y` for all :math:`y ∈ Y`, and such that :math:`\| T̄ \|_X  = \| T \|_Y`, where :math:`\| T̄ \|_X` and :math:`\| T \|_Y` are the usual operator :term:`norms <norm>`,

      .. math:: \|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 :term:`Compactness theorem`.

    Hellinger-Toeplitz theorem
      Every everywhere-defined :term:`self-adjoint` :term:`linear operator` on a :term:`Hilbert space` is :term:`bounded <bounded linear transformation>`.

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

    Hölder's inequality
      Let :math:`p` and :math:`q` be :term:`conjugate exponents` and :math:`(X, 𝔐, μ)` a measure space. 
      
      Assume :math:`f, g ≥ 0` are nonnegative, :term:`measurable functions <measurable function>` on :math:`X` with range in :math:`[0,∞]`.

      #. If :math:`1 < p < ∞`, then

         .. math:: ∫_X fg \, dμ ≤ \left( ∫_X f^p \, dμ\right)^{1/p} \left(∫_X g^q \, dμ\right)^{1/q}, \text{ and}
            :label: holder'

      #. If :math:`p = ∞`, :math:`f ∈ L_∞`, and :math:`g ∈ L_1`, then :math:`|f g| ≤ \|f\|_∞ |g|`, so :math:`\|f g\|_1 ≤ \|f\|_∞ \|g\|_1`.

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

      (See also :term:`Minkowski's inequality`.)
      
    integrability of a measurable function
      Let :math:`f: X → [-∞, ∞]` be a :term:`measurable function` on :math:`X`. Then the :term:`positive <positive part>` and :term:`negative <negative part>` parts of :math:`f` (denoted by :math:`f^+` and :math:`f^-`, respectively) are :term:`integrable` over :math:`X` if and only if :math:`|f|` is integrable over :math:`X`.

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           *Proof*.

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

    integral extensionality
      Suppose :math:`f` and :math:`g` are :term:`integrable` functions such that for all :term:`measurable <measurable set>` :math:`E` we have :math:`∫_E f = ∫_E g`. Then :math:`f = g` :math:`μ`-a.e.

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

      #. there exist open sets :math:`U` and :math:`V` in :math:`ℝ^n` such that :math:`a ∈ U`, :math:`b ∈ V`, and :math:`f` maps :math:`U` bijectively onto :math:`V`, and
      #. if :math:`g` is the inverse of :math:`f` (which exists by (i)), defined on :math:`V` by :math:`g(f(x))=x`, for :math:`x ∈ U`, then :math:`g ∈ C^1(V)`.
  
      See also Rudin, *Principles of Mathematical Analysis* :cite:`Rudin:1976`.

    inverse mapping theorem
      Let :math:`X, Y` be :term:`Banach spaces <Banach space>`. A continuous bijection :math:`T: X → Y` has a continuous inverse.

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

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

    mean value theorem
      If a real-valued function :math:`f` is :term:`continuous <continuous function>` on the (closed, bounded) interval :math:`[a, b]` and differentiable on :math:`(a, b)`, then there exists :math:`c ∈ (a,b)` such that
      
      .. math:: f'(c) = \frac{f(b) - f(a)}{b-a}.

    mean value theorem (version 2)
      If a real-valued function :math:`f` is :term:`continuous <continuous function>` on the :math:`closed <closed set>` :term:`bounded <bounded set>` interval :math:`[c, d]` and differentiable on its :term:`interior` :math:`(c, d)` with :math:`f' > α` on :math:`(c, d)`, then :math:`α ⋅ (d - c) < f(d)-f(c)`.

    measurability of upper limit
      If :math:`f_n: X → [-∞,∞]` is :term:`measurable <measurable function>` for each :math:`n∈ ℕ`, :math:`g = \sup\limits_{n ≥ 0} f_n`, and :math:`h = \limsup\limits_{n→ ∞} f_n`, then :math:`g` and :math:`h` are measurable.

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           *Proof*.

      :math:`g^{-1}((α, ∞]) = ⋃_{n∈ ℕ} f^{-1}((α, ∞])`, so :math:`g` is measurable. The same result holds with inf in place of sup, and since
      
      .. math:: h = \inf\limits_{k≥ 0} \left\{ \sup\limits_{i≥ k} f_i\right\},

      it follows that :math:`h` is measurable.

    Minkowski's inequality
      Let :math:`1 < p < ∞` and :math:`q` be :term:`conjugate exponents` and let :math:`(X, 𝔐, μ)` be a :term:`measure space`. If :math:`f, g` are :term:`measurable functions <measurable function>` on :math:`X`, then

         .. math:: \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 :term:`Hölder's inequality`.)

    monotone convergence theorem
      Let :math:`\{f_n\}` be a sequence of measurable functions on :math:`X`, and suppose that, for every :math:`x ∈ X`,

      #. :math:`0 ≤ f_1(x) ≤ f_2(x) ≤ \cdots ≤ ∞`,
      #. :math:`f_n(x) → f(x)` as :math:`n → ∞`.

      Then :math:`f` is measurable, and :math:`∫_X f_n \, dμ → ∫_X f \, dμ` as :math:`n → ∞`.

    open mapping theorem
      A surjective :term:`bounded linear transformation` from one :term:`Banach space` *onto* another is an :term:`open mapping`.

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

      #. If :math:`0 ≤ f ≤ g`, then :math:`∫_E f ≤ ∫_E g`.
      #. If :math:`f≥ 0`, then :math:`∫_A f ≤ ∫_B f`.
      #. If :math:`f≥ 0`, then :math:`∫_E cf = c∫_E f`.
      #. If :math:`f(x) = 0` for every :math:`x ∈ E`, then :math:`∫_E f = 0`, even if :math:`E` has infinite measure.
      #. If :math:`E` is :term:`negligible`, then :math:`∫_E f = 0`, even if :math:`f(x) = ∞` for all :math:`x ∈ E`.
      #. If :math:`f≥ 0`, then :math:`∫_E f = ∫_X χ_E  f`.

    properties of measures
      Let :math:`(X, 𝔐, μ)` be a :term:`measure space`. Then

      #. :math:`μ(A_1 ∪ \cdots ∪ A_n) = μ A_1 + \cdots + μ A_n` if :math:`A_1, \dots, A_n` are pairwise disjoint sets in :math:`𝔐`;
      #. :math:`μ ∅ = 0`;
      #. :math:`μ A ≤ μ B` if :math:`A ⊆ B` and :math:`A, B ∈ 𝔐`;
      #. :math:`μ A_n → μ A` as :math:`n→ ∞` if :math:`∀ n, A_n ∈ 𝔐` and :math:`A_1 ⊆ A_2 ⊆ \cdots` and :math:`A = ⋃_n A_n`.
      #. :math:`μ A_n → μ A` as :math:`n→ ∞` if :math:`∀ n, A_n ∈ 𝔐` and :math:`A_1 ⊇ A_2 ⊇ \cdots` and :math:`A = ⋂_n A_n` and :math:`μ A_1 < ∞`.

    Radon-Nikodym theorem
      If :math:`λ` and :math:`m` are :term:`σ-finite <σ-finite measure>` positive :term:`measures <measure>` on a :term:`σ-algebra` :math:`Σ` and if :math:`λ ≪ m`, then there is a unique :math:`g ∈ L_1(dm)` such that :math:`∀ E ∈ Σ`, :math:`λ E = ∫_E g \, dm`.

    Radon-Nikodym theorem (full version)
      Let :math:`(X, 𝔐, μ)` be a :term:`measure space` and assume :math:`μ` is a positive :term:`σ-finite measure`.
      
      If :math:`λ` is a :term:`complex measure` on :math:`𝔐`, then

      #. there is then a unique pair of complex measures :math:`λ_a` and :math:`λ_s` on :math:`𝔐` such that

         .. math:: λ = λ_a + λ_s,  \quad λ_a ≪ μ, \quad λ_s ⟂  μ;

         if :math:`λ` happens to be a real positive :term:`finite measure`, then so are :math:`\lambda_a` and :math:`\lambda_s`;

      #. there is a unique :math:`h ∈ L_1(μ)` such that

         .. math:: λ_a E = ∫_E h \, dμ \quad ∀ E ∈ 𝔐.

      The pair :math:`(λ_a, λ_s)` is called the **Lebesgue decomposition** of :math:`λ` relative to :math:`μ`.

    Radon-Nikodym corollary
      Suppose :math:`ν` is a :term:`σ-finite <σ-finite measure>` :term:`complex measure` and :math:`μ, λ` are :math:`σ`-finite measures on :math:`(X, 𝔐)` such that :math:`ν ≪ μ ≪ λ`. Then

      #. If :math:`g ∈ L_1(ν)`, then :math:`g \frac{dν}{dμ} ∈ L_1(μ)` and

         .. math:: ∫ g\, dν = ∫ g \frac{dν}{dμ} \, dμ.

      #. :math:`ν ≪ λ`, and

         .. math:: \frac{dν}{dλ} = \frac{dν}{dμ} \frac{dμ}{dλ} \quad λ\text{-a.e.}

    Riesz representation theorem
      Let :math:`X` be a :term:`locally compact` :term:`Hausdorff space`, and let :math:`Λ` be a positive :term:`linear functional` on :math:`C_c(X)`.
      
      There exists a :term:`σ-algebra` :math:`𝔐` in :math:`X` that contains all :term:`Borel sets <Borel set>` in :math:`X`, and there exists a unique positive measure :math:`μ` on :math:`𝔐` that represents :math:`Λ` in the following sense:

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

      #. :math:`μ K < ∞` for every compact set :math:`K ⊆ X`;

      #. for every :math:`E ∈ 𝔐`, we have

         .. math:: μ E = \inf \{ μ V ∣ E ⊆ V, V \text{ open}\};

      #. the relation

         .. math:: μ E = ∑ \{ μ K ∣ K ⊆ E, K \text{ compact}\}

         holds for every open set :math:`E`, and for every :math:`E ∈ 𝔐` with :math:`μ E < ∞`;

      #. if :math:`E ∈ 𝔐`, :math:`A ⊆ E`, and :math:`μ E = 0`, the :math:`A ∈ 𝔐`.

      (See also Rudin, *Real and Complex Analysis* :cite:`Rudin:1987` 2.14.) 

    Riesz representation theorem (version 2)
      Suppose :math:`1 < p < ∞` and :math:`\frac{1}{p} + \frac{1}{q} = 1`. If :math:`Λ` is a :term:`linear functional` on :math:`L_p`, then there is a unique :math:`g ∈ L_q` such that :math:`Λ f = ∫ f g \, dμ` for all :math:`f ∈ L_p`.

    Stone-Weierstrass theorem
      Let :math:`X` be a :term:`compact <compact set>` :term:`Hausdorff space` and let :math:`𝔄` be a subalgebra of :math:`C(X,ℝ)` that :term:`separates the points` of :math:`X` and contains the constant functions.  Then :math:`𝔄` is dense in :math:`C(X, ℝ)`.

    Stone-Weierstrass theorem (version 2)
      Let :math:`X` be a :term:`compact <compact set>` :term:`Hausdorff space` and let :math:`𝒜` be a closed subalgebra of functions in :math:`C(X,ℝ)` that :term:`separates the points` of :math:`X`. Then either :math:`𝒜 = C(X,ℝ)`, or :math:`𝒜 = \{f ∈ C(X,ℝ) ∣ f(x_0) = 0\}` for some :math:`x_0 ∈ X`. The first case occurs iff :math:`𝒜` contains the constant functions.

    Tonelli's theorem
      Let :math:`(X, 𝔄, μ)` and :math:`(Y, 𝔅, ν)` be :term:`σ-finite <σ-finite measure>` :term:`complete measure spaces <complete measure space>` and let :math:`f` be a nonnegative :math:`(μ × ν)`-:term:`measurable function` on :math:`X × Y`. Then,

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

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

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

         .. math:: ∫_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 :term:`Fubini's theorem`. [1]_

    Tychonoff's theorem
      Let :math:`{X_α}_{α∈𝔄}` be a be a collection of :term:`compact <compact set>` :term:`topological spaces <topological space>` indexed by a set :math:`𝔄`. Then the :term:`Cartesian product` :math:`∏_{α∈𝔄}X_α` with the :term:`product topology` also is compact.

    uniform boundedness principle
      See the :term:`Banach-Steinhaus theorem`.

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.. rubric:: Footnotes

.. [1]
   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* :cite:`Rudin:1987`.  Rudin begins by assuming only that the function :math:`f(x,y)` is measurable with respect to the :term:`product σ-algebra` :math:`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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.. include:: hyperlink_references.rst