Ε-net

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The correct title of this article is ε-net. The initial letter is shown capitalized due to technical restrictions.

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Let P be a probability distribution over some set X. An ε-net for a class H \subseteq 2^X of subsets of X is any subset S \subseteq X such that for any h \in H

P(h) \ge \epsilon \quad \Longrightarrow \quad S\cap h \neq \emptyset

Intuitively S approximates the probability distribution.

Stronger notion is ε-approximation. An ε-approximation for class H is subset S \subseteq X such that for any h \in H it holds

\left| P(h) - \frac{|S \cap h|}{|S|} \right| < \epsilon
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