• Title: Carathéodory’s Extension Theorem

  • Series: Measure Theory

  • YouTube-Title: Measure Theory 12 | Carathéodory’s Extension Theorem

  • Bright video: https://youtu.be/dSys4Tg6By0

  • Dark video: https://youtu.be/Wpih_xOgsig

  • Quiz: Test your knowledge

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  • Subtitle on GitHub: mt12_sub_eng.srt missing

  • Other languages: German version

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  • Quiz Content

    Q1: What is not correct for a semiring of sets $\mathcal{A} \subseteq \mathcal{P}(X)$.

    A1: $A,B \in \mathcal{A} \Rightarrow A \cup B \in \mathcal{A}$.

    A2: $\emptyset \in \mathcal{A}$

    A3: $A,B \in \mathcal{A} \Rightarrow A \cap B \in \mathcal{A}$.

    A4: For $A,B \in \mathcal{A}$ one finds pairwise disjoint sets $S_j \in \mathcal{A}$ with $$ \bigcup_{j=1}^n S_j = A \setminus B$$

    Q2: What is not correct for a premeasure $\mu: \mathcal{A} \rightarrow [0, \infty]$?

    A1: $\mu(A \cap B) = \mu(A) - \mu(B)$.

    A2: $\mathcal{A}$ is a semiring of sets.

    A3: $\mu(\emptyset) = 0$

    A4: $\mu(A \cup B) = \mu(A) + \mu(B)$ for $A,B \in \mathcal{A}$ disjoint where also $A \cup B \in \mathcal{A}$.

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