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Title: Image measure and substitution rule
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Series: Measure Theory
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YouTube-Title: Measure Theory 15 | Image measure and substitution rule
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Bright video: https://youtu.be/q3UgXso-1jw
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Dark video: https://youtu.be/xYnNTtIF9oM
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Ad-free video: Watch Vimeo video
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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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Print-PDF: Download printable PDF version
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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: mt15_sub_eng.srt missing
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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 $X$. How can we define a measure $\widetilde{\mu}$ on $Y$?
A1: $\widetilde{\mu}(B) = \mu( h^{-1} [ B ] )$
A2: $\widetilde{\mu}(B) = \mu( h [ B ] )$
A3: $\widetilde{\mu}(B) = \mu( B )$
A4: $\widetilde{\mu}(B) = h^{-1} [ B ] $
Q2: 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 $Y$. What is the correct substitution formula?
A1: $\int_Y g , d(h_{\ast} \mu) = \int_X g \circ h , d \mu$
A2: $\int_X g , d(h_{\ast} \mu) = \int_Y g \circ h , d \mu$
A3: $\int_Y g , d \mu = \int_X g \circ h , d(h_{\ast} \mu)$
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Last update: 2024-10
