Measurable functions and sets

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Problem: Show that any measurable function on \( \mathbb{R} \) (with the standard Lebesgue measure) is equivalent to some Borel function.

19.6.2.1 Measurable functions and sets

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Problem: Show that for any measurable function \( f \) on the segment \( [0,1] \) with Lebesgue measure and any \( \varepsilon>0 \) there exists such \( C \) with \( \mu(C)>1-\varepsilon \) and a continuous function \( g \) on the segment, that \( f(x)= \) \( g(x) \) when \( x \in C \).

19.6.2.2 Measurable functions and sets

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Problem: Let \( f(x) \geq 0 \) for \( \mu \) - nearly all \( x \in A \) and \[ \int_{A} f(x) \mu(d x)=0 . \] Prove that \( f(x)=0 \mu \)-almost everywhere on \( A \).

19.6.2.3 Measurable functions and sets

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Problem: What is the cardinality and measure of the set of points of the interval \( (0 ; 1) \) in a hexadecimal notation of which there are no digits 2 and 4.

19.6.2.4 Measurable functions and sets

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Problem: Prove that for any set \( A \subset \mathbb{R} \), such a measurable set \( B \) will be found, that \[ A \subset B \text { and } \mu^{*}(A)=\mu^{*}(B) \text {. } \]

19.6.2.5 Measurable functions and sets

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Problem: Let \( A \subset(0 ; 1) \). Let's suppose that \( f(x)=\mu^{*}(A \cap \) \( (\sqrt[5]{x} ; 1)) \). Prove that the function \( f(x) \) decreases, and is continuous on the segment \( [0 ; 1] \).

19.6.2.6 Measurable functions and sets

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Problem: Let \( E \) is a non-measurable set on the line, and \( A \) is a null set on the line. Prove that \( E \backslash A \) is non-measurable.

19.6.2.7 Measurable functions and sets

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Problem: Let \( A_{1}, A_{2} \) be measurable subsequences \( [0 ; 1] \) and \( \mu^{*}\left(A_{1}\right)+\mu^{*}\left(A_{2}\right)>1 \). Prove that \( \mu^{*}\left(A_{1} \cap A_{2}\right)>0 \).

19.6.2.8 Measurable functions and sets

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Problem: Prove that the set \( A \) will be measurable if for any number \( \varepsilon>0 \) such an open set \( B_{\varepsilon} \) will be found, that \( A \subset B_{\varepsilon} \) and \( \mu^{*}\left(B_{\varepsilon} \backslash A\right)<\varepsilon \).

19.6.2.9 Measurable functions and sets

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Problem: Let \( A \) be a measurable subsequence of the segment \( [0 ; 1] \), which has the following property: the distance between any two points from \( A \) is an irrational number. Prove that \( \mu^{*}(A)=0 \).

19.6.2.10 Measurable functions and sets

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Problem: Let \( f_{1}=g_{1} \) and \( f_{2}=g_{2} \) a. e. (almost everywhere) on the set \( A \). Prove that \( f_{1}+f_{2}=g_{1}+g_{2} \) a. e. on the set \( A \).

19.6.2.11 Measurable functions and sets

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Problem: Let \( f(x) \leq 0 \) a. e. (almost everywhere) on \( A \) and \( g(x) \leq \) \( \leq 0 \) a. e. on \( A \), let \( h(x)=\max (f(x), g(x)) \). Prove that \( h(x) \leq 0 \) a. e. on \( A \).

19.6.2.12 Measurable functions and sets

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