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1 Primary & Middle School Mathematics1–9 年级在线 2 High School Mathematics10–12 年级在线 3 University Mathematics — Year 1本科一年级在线 4 University Mathematics — Year 2本科二年级在线 5 University Mathematics — Year 3本科三年级在线

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1 Markets I: The Ecosystem and Exchange-Traded Markets市场在线 2 Markets II: Rates, FX and Credit市场在线 3 Markets III: Commodities, Energy and Crypto市场在线 4 Quantitative Methods方法在线 5 Derivatives and Volatility衍生品在线 6 Rates, Credit, XVA and Risk利率、信用与风险在线 7 Research Craft: Predictors, Backtests, Measurement, Portfolios研究在线 8 Strategies I: Equities and Futures策略在线 9 Strategies II: Volatility, Relative Value, Macro and the Bank Desks策略在线 10 Microstructure and Execution交易执行在线 11 Market Making and High-Frequency Trading做市在线 12 Machine Learning for Markets机器学习在线 13 Low-Latency Software技术在线 14 Networks, Hardware and Trading Infrastructure技术在线 15 Research, Data and Risk Platforms技术在线 16 The Desk and the Firm公司运营在线 17 The Industry: Firms, Roles and Careers职业发展在线 18 Interviews I: Maths, Probability and Speed职业发展在线 19 Interviews II: Markets, Trading Games and the Final Round职业发展在线 20 Interviews III: Code职业发展在线

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什么是 measure？

Definition 9.5 University Mathematics — Year 3 · 第 9 章 — Measure Theory

A measure on $(X, \mathcal A)$ is a map $\mu \colon \mathcal A \to [0, +\infty]$ with $\mu(\varnothing) = 0$ that is $\sigma$ -additive: for pairwise disjoint $(A_n)_{n\in\N}$ ,

\mu\Bigl(\bigsqcup_n A_n\Bigr) = \sum_n \mu(A_n).

$(X, \mathcal A, \mu)$ is a measure space; $\mu$ is finite if $\mu(X) < \infty$ , a probability measure if $\mu(X) = 1$ , $\sigma$ -finite if $X$ is a countable union of sets of finite measure. Examples: counting measure on $(\N, \mathcal P(\N))$ ; the Dirac mass $\delta_a(A) = \mathbf 1_{a \in A}$ ; and, the object of this chapter, Lebesgue measure.

示例

Example 9.14

The Cantor set (Exercise 6.10) has $\lambda(C) = 0$ : $C \subseteq C_n$ , a union of $2^n$ intervals of length $3^{-n}$ , so $\lambda(C) \leq (2/3)^n \to 0$ . An uncountable null set — cardinality does not see measure. Conversely, fat Cantor sets (Exercise 9.5) are nowhere dense with positive measure: topology does not see measure either. The weekend problem pushes this interplay to its striking conclusion: there are Lebesgue-measurable sets that are not Borel.

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