…情况下，有 Moore-Smith 收敛、网（nets）、滤子（filters）等等。在某些环节他还会发现连续统假设、模型论（model theory）等等。但无论如何，千万别让他碰 John C. Oxtoby 的《Measure and Category: A Survey of the Analogies between Topological and Measure Spaces》，ISBN 3-540-05349-2，Springer-Verlag, Berlin, 1971。要是他看到了这本书，恐怕就再也回不来了！
…case, there is Moore-Smith convergence, nets, filters, etc. Somewhere in there he will discover the continuum hypothesis, model theory, etc. But under no circumstances let him get near John C. Oxtoby, 'Measure and Category: A Survey of the Analogies between Topological and Measure Spaces', ISBN 3-540-05349-2, Springer-Verlag, Berlin, 1971. If he sees this book, he may never recover!
…分析与概率论课程。分析部分嘛，基本上就是 Royden 的《Real Analysis》加上 Rudin 的《Real and Complex Analysis》前半部分。还有一些选自 Oxtoby《Measure and Category》的内容（就像冰淇淋加蛋糕一样的甜点——超级好玩的内容）。概率论则是从一开始就讲西格玛代数（sigma algebras）等等。所以核心主题是 Radon-Nikodym 定理和条件期望——美妙极了…
…course in analysis and probability. So, the analysis part was, right, basically Royden, Real Analysis and the first half of Rudin, Real and Complex Analysis. There was also some material from Oxtoby, Measure and Category (ice cream and cake dessert -- super fun stuff). The probability was right from the beginning sigma algebras, etc. So, a central topic was the Radon-Nikodym theorem and conditional expectation -- gorgeous…
…关于上/下半连续性的习题，而且有一个比 Littlewood 三大原则更好的总结。Bernard R. Gelbaum and John M. H. Olmsted, Counterexamples in Analysis John C. Oxtoby, Measure and Category: A Survey of the Analogies between Topological and Measure Spaces Walter Rudin, Real and Complex Analysis Walter Rudin, Functional Analysis Leo Breiman, Probability Kai Lai Chung, A Course in Probability Theory,…
…exercises about upper/lower semi-continuity, and there is a better summary than Littlewood's three principles. Bernard R. Gelbaum and John M. H. Olmsted, Counterexamples in Analysis John C. Oxtoby, Measure and Category: A Survey of the Analogies between Topological and Measure Spaces Walter Rudin, Real and Complex Analysis Walter Rudin, Functional Analysis Leo Breiman, Probability Kai Lai Chung, A Course in Probability Theory,…
John Oxtoby 的《Measure and Category》。这本书研究了测度论和拓扑学中关于“小”集合不同概念之间的对偶结果。这是第一本（在某种程度上也是唯一一本）让我瞬间顿悟的数学书…
Measure and Category by John Oxtoby. This book studies duality results between different notions of "small" sets in measure theory and topology. It's the first (and to some extent the only) math book where things just clicked…
…例如在工程学的某些领域，这并不平凡，但在 Luenberger 的处理下变得通俗易懂。这给人上了一课：某些纯数学在应用中可能远比乍看之下强大得多。John C. Oxtoby 的《Measure and Category: A Survey of the Analogies between Topological and Measure Spaces》，优雅，震撼。通过 Baire 纲定理学到的一些东西甚至能颠覆一个人对实数线和我们所在三维空间的直观认知。…
…e.g., in parts of engineering, is not trivial and is made easy by Luenberger. It's a lesson: Some of pure math can be much more powerful in applications than is easy to see at first. John C. Oxtoby, 'Measure and Category: A Survey of the Analogies between Topological and Measure Spaces' Elegant. Astounding. Some of what learn via the Baire category theorem can shake one's intuitive view of the real line and our 3-space.…
…更一般。然后注意 Carathéodory 扩张定理——这是在实数线上定义测度（尤其是 Lebesgue 测度）的关键。至少花一个周末读读 John C. Oxtoby 的《Measure and Category: A Survey of the Analogies between Topological and Measure Spaces》，ISBN 3-540-05349-2，Springer-Verlag, Berlin, 1971。这会是一个极其美妙的周末。手头常备 Bernard R. Gelbaum 和 John M. H. Olmsted 的…
…more general. Then notice the Carathéodory extension result -- that is the key to defining measures on the real line, in particular, Lebesgue measure. Spend at least a weekend in John C.\ Oxtoby, {\it Measure and Category:\ \ A Survey of the Analogies between Topological and Measure Spaces,\/} ISBN 3-540-05349-2, Springer-Verlag, Berlin, 1971.\ \ An amazing weekend. Keep at hand Bernard R.\ Gelbaum and John M.\ H.\ Olmsted,…
…情况下，有 Moore-Smith 收敛、网（nets）、滤子（filters）等等。在某些环节他还会发现连续统假设、模型论（model theory）等等。但无论如何，千万别让他碰 John C. Oxtoby 的《Measure and Category: A Survey of the Analogies between Topological and Measure Spaces》，ISBN 3-540-05349-2，Springer-Verlag, Berlin, 1971。要是他看到了这本书，恐怕就再也回不来了！
…case, there is Moore-Smith convergence, nets, filters, etc. Somewhere in there he will discover the continuum hypothesis, model theory, etc. But under no circumstances let him get near John C. Oxtoby, 'Measure and Category: A Survey of the Analogies between Topological and Measure Spaces', ISBN 3-540-05349-2, Springer-Verlag, Berlin, 1971. If he sees this book, he may never recover!
…分析与概率论课程。分析部分嘛，基本上就是 Royden 的《Real Analysis》加上 Rudin 的《Real and Complex Analysis》前半部分。还有一些选自 Oxtoby《Measure and Category》的内容（就像冰淇淋加蛋糕一样的甜点——超级好玩的内容）。概率论则是从一开始就讲西格玛代数（sigma algebras）等等。所以核心主题是 Radon-Nikodym 定理和条件期望——美妙极了…
…course in analysis and probability. So, the analysis part was, right, basically Royden, Real Analysis and the first half of Rudin, Real and Complex Analysis. There was also some material from Oxtoby, Measure and Category (ice cream and cake dessert -- super fun stuff). The probability was right from the beginning sigma algebras, etc. So, a central topic was the Radon-Nikodym theorem and conditional expectation -- gorgeous…
…关于上/下半连续性的习题，而且有一个比 Littlewood 三大原则更好的总结。Bernard R. Gelbaum and John M. H. Olmsted, Counterexamples in Analysis John C. Oxtoby, Measure and Category: A Survey of the Analogies between Topological and Measure Spaces Walter Rudin, Real and Complex Analysis Walter Rudin, Functional Analysis Leo Breiman, Probability Kai Lai Chung, A Course in Probability Theory,…
…exercises about upper/lower semi-continuity, and there is a better summary than Littlewood's three principles. Bernard R. Gelbaum and John M. H. Olmsted, Counterexamples in Analysis John C. Oxtoby, Measure and Category: A Survey of the Analogies between Topological and Measure Spaces Walter Rudin, Real and Complex Analysis Walter Rudin, Functional Analysis Leo Breiman, Probability Kai Lai Chung, A Course in Probability Theory,…
John Oxtoby 的《Measure and Category》。这本书研究了测度论和拓扑学中关于“小”集合不同概念之间的对偶结果。这是第一本（在某种程度上也是唯一一本）让我瞬间顿悟的数学书…
Measure and Category by John Oxtoby. This book studies duality results between different notions of "small" sets in measure theory and topology. It's the first (and to some extent the only) math book where things just clicked…
…例如在工程学的某些领域，这并不平凡，但在 Luenberger 的处理下变得通俗易懂。这给人上了一课：某些纯数学在应用中可能远比乍看之下强大得多。John C. Oxtoby 的《Measure and Category: A Survey of the Analogies between Topological and Measure Spaces》，优雅，震撼。通过 Baire 纲定理学到的一些东西甚至能颠覆一个人对实数线和我们所在三维空间的直观认知。…
…e.g., in parts of engineering, is not trivial and is made easy by Luenberger. It's a lesson: Some of pure math can be much more powerful in applications than is easy to see at first. John C. Oxtoby, 'Measure and Category: A Survey of the Analogies between Topological and Measure Spaces' Elegant. Astounding. Some of what learn via the Baire category theorem can shake one's intuitive view of the real line and our 3-space.…
…更一般。然后注意 Carathéodory 扩张定理——这是在实数线上定义测度（尤其是 Lebesgue 测度）的关键。至少花一个周末读读 John C. Oxtoby 的《Measure and Category: A Survey of the Analogies between Topological and Measure Spaces》，ISBN 3-540-05349-2，Springer-Verlag, Berlin, 1971。这会是一个极其美妙的周末。手头常备 Bernard R. Gelbaum 和 John M. H. Olmsted 的…
…more general. Then notice the Carathéodory extension result -- that is the key to defining measures on the real line, in particular, Lebesgue measure. Spend at least a weekend in John C.\ Oxtoby, {\it Measure and Category:\ \ A Survey of the Analogies between Topological and Measure Spaces,\/} ISBN 3-540-05349-2, Springer-Verlag, Berlin, 1971.\ \ An amazing weekend. Keep at hand Bernard R.\ Gelbaum and John M.\ H.\ Olmsted,…