2.2 Limits and Decompositions of Sequences of Measurable Functions
5. Complex Measurable Functions and Extended Real-Valued Functions
For complex-valued functions \(f: X \to \mathbb{C}\), we utilize the product space topology of \(\mathbb{R}^2\) for the complex plane, namely \(\mathcal{B}_{\mathbb{C}} = \mathcal{B}_{\mathbb{R}} \otimes \mathcal{B}_{\mathbb{R}}\).
Proposition 2.1.4 (Complex Measurability)
A complex-valued function \(f\) is measurable if and only if its real part \(\text{Re } f\) and imaginary part \(\text{Im } f\) are both real measurable functions.
In measure theory, we often allow functions to take values of \(\pm \infty\). Define the extended real number set \(\bar{\mathbb{R}} = \mathbb{R} \cup \{-\infty, +\infty\}\), whose Borel \(\sigma\)-algebra is defined as:
่ฅ \(f\) ไธบๅนฟไนๅฎๅฏๆตๅฝๆฐ๏ผๆไปฌ้ๅธธ็บฆๅฎ \(0 \cdot \infty = 0\) ไธ้ฟๅ ๆ ๆไน็ \(\infty - \infty\)ใ
6. Algebraic Operations and Limits of Sequences
Measurable functions possess perfect closure under various operations and limit-taking procedures.
Theorem 2.1.3 (Closure under Algebraic Operations)
Let \(f, g: X \to \mathbb{C}\) be measurable functions, then \(f+g\) and \(f \cdot g\) are both measurable functions.
Proof of Theorem 2.1.3
We define the mapping \(F(x) = (f(x), g(x)): X \to \mathbb{C} \times \mathbb{C}\). Since \(f\) and \(g\) are both measurable, according to the measurability of product spaces (Proposition 2.1.3), the joint mapping \(F\) is measurable. Furthermore, since the complex addition mapping \(S: \mathbb{C} \times \mathbb{C} \to \mathbb{C}\), i.e., \(S(z, w) = z + w\), is a continuous mapping, it is therefore Borel measurable. By the composition property, \(f+g = S \circ F\) must be measurable. The measurability of the product \(f \cdot g\) can be proved analogously.
Theorem 2.1.4 (Supremum/Infimum and Limits of Function Sequences)
Let \(\{f_j\}_{j=1}^\infty\) be a sequence of extended real-valued measurable functions defined on \(X\), then the following functions are also measurable:
- \(g_1(x) = \sup_{j} f_j(x)\)
- \(g_2(x) = \inf_{j} f_j(x)\)
- \(g_3(x) = \limsup_{j \to \infty} f_j(x)\)
- \(g_4(x) = \liminf_{j \to \infty} f_j(x)\)
Proof of Theorem 2.1.4
We only prove the measurability of the supremum \(\sup_j f_j\); the rest can be derived from algebraic relations. For any \(a \in \mathbb{R}\), observe the set \(\{x : \sup_j f_j(x) > a\}\). The supremum is greater than \(a\) if and only if there exists at least one term in the sequence greater than \(a\). Therefore:
Since each \(f_j\) is measurable, every set \(\{x : f_j(x) > a\}\) on the right side of the equation is in \(\mathcal{M}\). Because \(\mathcal{M}\) is closed under countable unions, the supremum function \(g_1\) satisfies the equivalent condition for measurability (Proposition 2.1.2(b)), and is thus measurable.
Remaining proof: \(\inf_j f_j = -\sup_j (-f_j)\), which follows from algebraic operations and the measurability of the \(\sup\). Since \(\limsup_{j \to \infty} f_j = \inf_{k \ge 1} (\sup_{j \ge k} f_j)\), the measurability is naturally preserved after one countable \(\sup\) and one countable \(\inf\) operation. The same applies to \(\liminf\).
Corollary (Positive/Negative Parts and Polar Decomposition)
For any real measurable function \(f\), we can decompose it into positive and negative parts:
- Positive part: \(f^+ = \max(f, 0)\)
- Negative part: \(f^- = \max(-f, 0)\)
- At this point, we have \(f = f^+ - f^-\) and \(|f| = f^+ + f^-\). These two functions are also measurable.
For complex measurable functions, we can perform polar decomposition (sign decomposition):
where \(\text{sgn } z = z/|z|\) when \(z \ne 0\), and 0 otherwise. Both \(\text{sgn } f\) and \(|f|\) are also measurable functions.