2.1 Concepts and Properties of Measurable Functions
In a measure space \((X, \mathcal{M})\), measurable functions are the central objects in the study of integration theory. Just as continuous functions preserve the structure of open sets in topological spaces, measurable functions preserve the Ο-algebra structure between measurable spaces.
1. Properties of Mappings and Inverse Images
Before defining measurable functions, we recall the basic properties of inverse images of sets. The inverse image mapping \(f^{-1}: 2^Y \to 2^X\) is completely compatible with various set operations.
Lemma 2.1.1 (Properties of Inverse Images)
Let \(f: X \to Y\) be a mapping and \(E, E_{\lambda} \subset Y\). Then:
- Preservation of unions:
- Preservation of intersections:
- Preservation of complements:
From these properties it follows that if \(\mathcal{N}\) is a Ο-algebra on \(Y\), then the collection \(\mathcal{F} = \{E \subset Y : f^{-1}(E) \in \mathcal{M}\}\) forms a Ο-algebra on \(Y\).
2. Definition and Basic Criterion of Measurable Functions
Definition 2.1.1 (\((\mathcal{M}, \mathcal{N})\)-Measurability)
Let \((X, \mathcal{M})\) and \((Y, \mathcal{N})\) be two measurable spaces. A mapping \(f: X \to Y\) is called \((\mathcal{M}, \mathcal{N})\)-measurable (or simply measurable) if for every \(E \in \mathcal{N}\), its inverse image satisfies
When the target space is the real line \(\mathbb{R}\) equipped with the Borel Ο-algebra \(\mathcal{B}_{\mathbb{R}}\), we call \(f\) an \(\mathcal{M}\)-measurable function.
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Borel measurable function: A function \(f: (\mathbb{R}, \mathcal{B}_{\mathbb{R}}) \to (\mathbb{R}, \mathcal{B}_{\mathbb{R}})\) that is measurable.
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Lebesgue measurable function: A function \(f: (\mathbb{R}, \mathcal{L}) \to (\mathbb{R}, \mathcal{B}_{\mathbb{R}})\) that is measurable.
Proposition 2.1.1 (Criterion via Generators)
If the Ο-algebra \(\mathcal{N}\) on the target space is generated by a family \(\Sigma\) (i.e., \(\mathcal{N} = \mathcal{M}(\Sigma)\)), then \(f\) is \((\mathcal{M}, \mathcal{N})\)-measurable if and only if
Proof of Proposition 2.1.1
Necessity: If \(f\) is measurable, then by definition \(f^{-1}(E) \in \mathcal{M}\) for all \(E \in \mathcal{N}\). Since \(\Sigma \subset \mathcal{N}\), this holds for all \(E \in \Sigma\).
Sufficiency: Consider the family \(\mathcal{F} = \{E \subset Y : f^{-1}(E) \in \mathcal{M}\}\). By Lemma 2.1.1, because the inverse image commutes with unions, intersections, and complements, \(\mathcal{F}\) is a Ο-algebra on \(Y\). By hypothesis, \(\Sigma \subset \mathcal{F}\). Since \(\mathcal{N}\) is the smallest Ο-algebra containing \(\Sigma\), we must have \(\mathcal{N} = \mathcal{M}(\Sigma) \subset \mathcal{F}\). This means that for every \(E \in \mathcal{N}\), \(f^{-1}(E) \in \mathcal{M}\); hence \(f\) is measurable.
Corollary (Measurability of Continuous Maps)
If \(X, Y\) are topological spaces and \(f: X \to Y\) is continuous, then \(f\) is \((\mathcal{B}_X, \mathcal{B}_Y)\)-measurable (i.e., Borel measurable).
Proof of the Corollary
The Borel Ο-algebra \(\mathcal{B}_Y\) on the topological space \(Y\) is generated by the family \(\mathcal{T}_Y\) of all open sets in \(Y\). Take any open set \(V \in \mathcal{T}_Y\). Since \(f\) is continuous, the inverse image \(f^{-1}(V)\) is an open set in \(X\). Open sets in \(X\) certainly belong to its Borel Ο-algebra \(\mathcal{B}_X\). By Proposition 2.1.1, a continuous function is necessarily Borel measurable.
3. Equivalent Conditions for Real-Valued Measurable Functions
For functions defined on the real line, because the Borel Ο-algebra can be generated by various forms of rays, we have the following commonly used equivalent criteria.
Proposition 2.1.2 (Equivalent Conditions for Real-Valued Measurable Functions)
For a mapping \(f: X \to \mathbb{R}\), the following statements are equivalent:
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(a) \(f\) is measurable (i.e., \(f^{-1}(B) \in \mathcal{M},\ \forall B \in \mathcal{B}_{\mathbb{R}}\)).
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(b) For every \(a \in \mathbb{R}\), \(f^{-1}((a, +\infty)) = \{x \in X : f(x) > a\} \in \mathcal{M}\).
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(c) For every \(a \in \mathbb{R}\), \(f^{-1}([a, +\infty)) = \{x \in X : f(x) \ge a\} \in \mathcal{M}\).
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(d) For every \(a \in \mathbb{R}\), \(f^{-1}((-\infty, a)) = \{x \in X : f(x) < a\} \in \mathcal{M}\).
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(e) For every \(a \in \mathbb{R}\), \(f^{-1}((-\infty, a]) = \{x \in X : f(x) \le a\} \in \mathcal{M}\).
Proof of Proposition 2.1.2
Because the Borel Ο-algebra \(\mathcal{B}_{\mathbb{R}}\) on \(\mathbb{R}\) is generated by the family of open rays \(\Sigma = \{(a, +\infty) : a \in \mathbb{R}\}\), Proposition 2.1.1 tells us that (a) and (b) are equivalent. It remains to show that (b), (c), (d), (e) can be deduced from each other using the closure properties of a Ο-algebra:
- (b) \(\Rightarrow\) (c): Since \([a, +\infty) = \bigcap_{n=1}^\infty (a - \frac{1}{n}, +\infty)\), taking inverse images gives
By (b) each set on the right belongs to \(\mathcal{M}\); because a Ο-algebra is closed under countable intersections, the left-hand side is also in \(\mathcal{M}\).
- (c) \(\Rightarrow\) (d): Since \((-\infty, a) = ([a, +\infty))^c\), taking inverse images yields
By (c), \(f^{-1}([a, +\infty)) \in \mathcal{M}\), and closure under complements gives that the inverse image belongs to \(\mathcal{M}\).
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(d) \(\Rightarrow\) (e): Similar to above, use \((-\infty, a] = \bigcap_{n=1}^\infty (-\infty, a + \frac{1}{n})\) and take inverse images.
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(e) \(\Rightarrow\) (b): Use \((a, +\infty) = ((-\infty, a])^c\) and take inverse images and complements, completing the circular argument.
4. Local Measurability, Composition, and Product Spaces
Definition 2.1.2 (Local Measurability)
Let \(E \in \mathcal{M}\) be a measurable set. A function \(f\) is said to be measurable on \(E\) if for every \(B \in \mathcal{B}_{\mathbb{R}}\),
This is equivalent to saying that the restriction \(f|_E\) is measurable with respect to the subβΟ-algebra \(\mathcal{M}_E = \{A \cap E : A \in \mathcal{M}\}\).
Theorem 2.1.2 (Composition Property)
If \(f: X \to Y\) is \((\mathcal{M}, \mathcal{N})\)-measurable and \(g: Y \to Z\) is \((\mathcal{N}, \mathcal{P})\)-measurable, then the composition \(g \circ f: X \to Z\) is \((\mathcal{M}, \mathcal{P})\)-measurable.
Proof of Theorem 2.1.2
For any \(E \in \mathcal{P}\), because \(g\) is \((\mathcal{N}, \mathcal{P})\)-measurable, we have \(g^{-1}(E) \in \mathcal{N}\). Since \(f\) is \((\mathcal{M}, \mathcal{N})\)-measurable, it follows that \(f^{-1}(g^{-1}(E)) \in \mathcal{M}\). Clearly \((g \circ f)^{-1}(E) = f^{-1}(g^{-1}(E)) \in \mathcal{M}\), which completes the proof. In particular, the composition of a measurable function with a continuous function is again measurable.
For a product space \(Y = \prod_{\alpha \in A} Y_{\alpha}\) equipped with the product Ο-algebra \(\mathcal{N} = \bigotimes_{\alpha \in A} \mathcal{N}_{\alpha}\), let \(\pi_\alpha : Y \to Y_\alpha\) be the projection mapping.
Proposition 2.1.3 (Measurability into Product Spaces)
A mapping \(f: X \to Y\) is \((\mathcal{M}, \mathcal{N})\)-measurable if and only if each of its coordinate mappings \(f_{\alpha} = \pi_{\alpha} \circ f\) is \((\mathcal{M}, \mathcal{N}_{\alpha})\)-measurable.
Proof of Proposition 2.1.3
Necessity (\(\Rightarrow\)): If \(f\) is measurable, then since the projection \(\pi_\alpha\) is always \((\mathcal{N}, \mathcal{N}_\alpha)\)-measurable, by the composition property \(f_\alpha = \pi_\alpha \circ f\) must be \((\mathcal{M}, \mathcal{N}_\alpha)\)-measurable.
Sufficiency (\(\Leftarrow\)): The product Ο-algebra \(\mathcal{N}\) is generated by the family \(\Sigma = \{\pi_\alpha^{-1}(E_\alpha) : E_\alpha \in \mathcal{N}_\alpha,\ \alpha \in A\}\). For any element of \(\Sigma\), its inverse image is
By hypothesis each \(f_\alpha\) is measurable, so the above set belongs to \(\mathcal{M}\). By the generator criterion (Proposition 2.1.1), \(f\) is measurable.