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Title: Proof of the substitution rule for measure spaces
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Series: Measure Theory
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Chapter: Construction of Measures
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YouTube-Title: Measure Theory 16 | Proof of the Substitution Rule for Measure Spaces
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Bright video: Watch on YouTube
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Forum: Ask a question in Mattermost
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Quiz: Test your knowledge
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Dark-PDF: Download PDF version of the dark video
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Subtitle on GitHub: mt16_sub_eng.srt missing
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Download bright video: Link on Vimeo
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Download dark video: Link on Vimeo
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Other languages: German version
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Timestamps (n/a)
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Subtitle in English (n/a)
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Quiz Content
Q1: Let’s consider two measurable spaces $(X, \mathcal{A})$, $(Y, \mathcal{B})$ and a measurable map $h: X \rightarrow Y$. We assume that $\mu$ is a measure on $X$ and $h_{\ast} \mu$ the image measure on $Y$. How does the substitution formula look for the function $\chi_C$, which only has the values $0$ and $1$?
A1: $ \displaystyle \int_Y \chi_C , d(h_{\ast} \mu) = \int_X \chi_C \circ h , d \mu$
A2: $ \displaystyle \int_X \chi_C , d(h_{\ast} \mu) = \int_Y \chi_C \circ h , d \mu$
A3: $ \displaystyle \int_Y \chi_C , d \mu = \int_X \chi_C \circ h , d(h_{\ast} \mu)$
Q2: If we know that the change of variables formula holds for every characteristic function $\chi_C$, how can we conclude that it also holds for every simple function?
A1: We use the linearity of the integral.
A2: We use the monotonicity of the integral.
A3: We use the fact that every integral is positive.
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Last update: 2025-11
