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30 Problem: Calculate the area of the domain \( D \), bounded by the given lines: \[ D:\left\{\begin{array}{c} y=\frac{1}{4} x^{2} \\ y=3 x-\frac{1}{2} x^{2} \end{array}\right. \]

9.7.8 The area of a region

1.02 $

30 Problem: Calculate the area of triangle \( A B C \) with the coordinates of vertices: \( A(2 ; 3), B(7 ; 2), C(4 ; 5) \).

9.7.9 The area of a region

1.27 $

3 Problem: Calculate the area of the domain \( D \), bounded by the given lines: \[ D:\left\{\begin{array}{l} y^{2}+2 y-3 x+1=0 \\ 3 x-3 y-7=0 \end{array}\right. \]

9.7.10 The area of a region

1.02 $

T Problem: Calculate the area of the domain \( D \), bounded by the given lines: \[ D: 2 x+3 y^{2}=0, \quad 2 x+2 y+1=0 . \]

9.7.11 The area of a region

1.27 $

3 Problem: Calculate the area of the domain \( D \), bounded by the given lines: \[ D: y=20-x^{2}, y=-8 x . \]

9.7.12 The area of a region

1.02 $

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

5.08 $

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

7.62 $

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

5.08 $

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

6.35 $

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

4.32 $

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

4.32 $

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

2.54 $

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

3.81 $

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

4.32 $

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

5.08 $

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