### 3.1. Uniform Law Under Finite Bracketing Entropy

In Chapter 3, we will focus on the uniform law of large numbers and its sufficiencies. The first part of the chapter consists of the simplest case: ULLN under finite bracketing entropy condition. After that, techniques to prove sufficiencies of ULLN will be introduced. Finally, ULLN under limiting $L^1$-entropy, popular function classes, or under constraints in VC dimension will be covered.

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### 2.3. Entropy Inequalities

This section covers the important results regarding the upper bound of entropies of popular classes of functions. We will accept most of the results without proving them, since it might obscure our objective: LLN and CLT on function spaces.

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### 2.2. Entropy

While computing complexity of classes of functions by its cardinality is natural, it reports complexity of every infinite dimensional classes to be the same. We need tools that can differentiate complexity more meticulously. Such tools are entropies of various kinds.

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### 2.1. Empirical Process

Chapter 2 concerns essential notions of the field. First, we will define the empirical distribution and the empirical process of a random sample. Next, various kinds of entropies and their basic relationships will be covered. Finally, upper bounds of entropy of some of the function classes will be mentioned.

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### 내 가장 오래된 프로그래밍 기억

초등학교 1 학년이었던가? 아무튼 저학년이었던 것은 확실하다. 컴퓨터 선생님이셨는지 담임 선생님이셨는지 아님 둘 다였는지 확실하게 기억나지 않는다. 아마 담임 선생님이셨던 것 같다. 어느날 선생님이 컴퓨터실에서 화면 상의 글자를 예쁘게 치장하는 방법을 알려주셨다. “HTML 태그”라는 것이었다.

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