……无界序列往往已经“足够长”，足以涵盖无限序列的所有有趣行为。0. Alain Robert 的《Nonstandard Analysis》是一个很好的起点。1. 参见他的短篇小册子《Radically Elementary Probability Theory》。我超爱这本书，在读它之前我并不怎么喜欢概率论。
…unlimited sequences often are "long enough" to incorporate all the interesting behavior of infinite sequences. 0. Alain Robert's Nonstandard Analysis is a good starting point. 1. See his short book Radically Elementary Probability Theory. I love this book, and didn't much like probability theory before reading it.
……与平滑函数做卷积（即卷一下）。但在我看来，这些托辞哪怕在研究生阶段都没有得到严谨的证实。* 除此之外，不知楼主有没有看过 Edward Nelson 的《Radically Elementary Probability Theory》（https://web.math.princeton.edu/~nelson/books/rept.pdf）。它利用非标准分析避开了测度论中大量（虽优雅但）极其繁琐的细节。光是前言读起来就很有意思。
…against a smooth function (i.e., a convolution) But these excuses are not rigorously justified, even at the graduate level, in my experience. * Separately from that, I wonder if OP has ever seen the book Radically Elementary Probability Theory, by Edward Nelson (https://web.math.princeton.edu/~nelson/books/rept.pdf). It uses nonstandard analysis to get around a lot of the (elegant) fussiness of measure theory. The preface alone is fun to read.
《Radically Elementary Probability Theory》（即上面的 REPT）是一本极好的书。非常薄的一册，任何硬啃过测度论概率的人都值得拿来至少翻一翻。它引人入胜且极具独创性，比如它的证明机制……
Radically Elementary Probability Theory (REPT above) is an excellent book. A very thin volume that anyone who labored through measure-theoretic probability would enjoy at least scanning. It is fascinating and original, like the proof mechanism…
……可行，我们不需要搬出西格玛代数（sigma-algebras）就能讨论条件概率或期望。在 Nelson 的书《Radically Elementary Probability Theory》中可以找到一种奇特怪诞的概率论替代实现——我强烈推荐读一读，借此看看底层框架到底有多无关紧要。
…works, and we don't need to talk about sigma-algebras to talk about conditional probabilities or expectations. A wacky alternative implementation of probability theory can be found in Nelson's book "Radically Elementary Probability Theory" - I highly recommend it as a way to see how little the underlying framework matters.
顺便提一句：任何对基于无穷小的微积分感兴趣的人，可能也会有兴趣看看 Ed Nelson 写的《Radically Elementary Probability Theory》：https://web.math.princeton.edu/~nelson/books/rept.pdf
Side comment: anyone interested in calculus via infinitesimals may also be interested in taking a look at Radically Elementary Probability Theory by Ed Nelson: https://web.math.princeton.edu/~nelson/books/rept.pdf
……在方法论上让我们教学的数字与我们实际测量和使用它们的方式保持一致。我个人一直在尝试针对大家熟悉的数学构建各种有限等价物。我建议大家读一读 Nelson 的《Radically Elementary Probability Theory》，至少在理论层面上更好地理解如何做有限数学。再说一次，在直接计算量的实际层面上，我们从来都只是在做有限数学。
…teaching about numbers methodologically aligned with how we actually measure and use them. I have personally been working on building various finite equivalents to familiar math. I recommend anyone to read Radically Elementary Probability Theory by Nelson to get a better sense of how to do finite math, at least at the theoretical level. Once again, on a practical level to do with directly computing quantities, we've only ever done finite math.
……无界序列往往已经“足够长”，足以涵盖无限序列的所有有趣行为。0. Alain Robert 的《Nonstandard Analysis》是一个很好的起点。1. 参见他的短篇小册子《Radically Elementary Probability Theory》。我超爱这本书，在读它之前我并不怎么喜欢概率论。
…unlimited sequences often are "long enough" to incorporate all the interesting behavior of infinite sequences. 0. Alain Robert's Nonstandard Analysis is a good starting point. 1. See his short book Radically Elementary Probability Theory. I love this book, and didn't much like probability theory before reading it.
……与平滑函数做卷积（即卷一下）。但在我看来，这些托辞哪怕在研究生阶段都没有得到严谨的证实。* 除此之外，不知楼主有没有看过 Edward Nelson 的《Radically Elementary Probability Theory》（https://web.math.princeton.edu/~nelson/books/rept.pdf）。它利用非标准分析避开了测度论中大量（虽优雅但）极其繁琐的细节。光是前言读起来就很有意思。
…against a smooth function (i.e., a convolution) But these excuses are not rigorously justified, even at the graduate level, in my experience. * Separately from that, I wonder if OP has ever seen the book Radically Elementary Probability Theory, by Edward Nelson (https://web.math.princeton.edu/~nelson/books/rept.pdf). It uses nonstandard analysis to get around a lot of the (elegant) fussiness of measure theory. The preface alone is fun to read.
《Radically Elementary Probability Theory》（即上面的 REPT）是一本极好的书。非常薄的一册，任何硬啃过测度论概率的人都值得拿来至少翻一翻。它引人入胜且极具独创性，比如它的证明机制……
Radically Elementary Probability Theory (REPT above) is an excellent book. A very thin volume that anyone who labored through measure-theoretic probability would enjoy at least scanning. It is fascinating and original, like the proof mechanism…
……可行，我们不需要搬出西格玛代数（sigma-algebras）就能讨论条件概率或期望。在 Nelson 的书《Radically Elementary Probability Theory》中可以找到一种奇特怪诞的概率论替代实现——我强烈推荐读一读，借此看看底层框架到底有多无关紧要。
…works, and we don't need to talk about sigma-algebras to talk about conditional probabilities or expectations. A wacky alternative implementation of probability theory can be found in Nelson's book "Radically Elementary Probability Theory" - I highly recommend it as a way to see how little the underlying framework matters.
顺便提一句：任何对基于无穷小的微积分感兴趣的人，可能也会有兴趣看看 Ed Nelson 写的《Radically Elementary Probability Theory》：https://web.math.princeton.edu/~nelson/books/rept.pdf
Side comment: anyone interested in calculus via infinitesimals may also be interested in taking a look at Radically Elementary Probability Theory by Ed Nelson: https://web.math.princeton.edu/~nelson/books/rept.pdf
……在方法论上让我们教学的数字与我们实际测量和使用它们的方式保持一致。我个人一直在尝试针对大家熟悉的数学构建各种有限等价物。我建议大家读一读 Nelson 的《Radically Elementary Probability Theory》，至少在理论层面上更好地理解如何做有限数学。再说一次，在直接计算量的实际层面上，我们从来都只是在做有限数学。
…teaching about numbers methodologically aligned with how we actually measure and use them. I have personally been working on building various finite equivalents to familiar math. I recommend anyone to read Radically Elementary Probability Theory by Nelson to get a better sense of how to do finite math, at least at the theoretical level. Once again, on a practical level to do with directly computing quantities, we've only ever done finite math.