Lebesgue measure and integration

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Problem: Let \( \mu(A)<\infty \). Prove that the non-negative function \( f \) is integrable with respect to \( A \) only when the following series converges: \[ \sum_{n=1}^{\infty} 2^{n} \mu\left(A \cap\left\{x: f(x) \geq 2^{n}\right\}\right) . \]

19.6.1.1 Lebesgue measure and integration

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Problem: Calculate the integral: \[ I=\int_{0}^{1} k(x) d k^{2}(x) \text {, } \] where \( k(x) \) is the Cantor function.

19.6.1.3 Lebesgue measure and integration

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Problem: The signed measure of \( v_{F} \), constructed from the function \( F(x)=\left\{\begin{array}{ll}e^{x}, & 0 \leq x \leq 2 \\ x^{2}, & 2

19.6.1.4 Lebesgue measure and integration

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Problem: 1) Find the Radon-Nikodym density with respect to the Lebesgue measure for the signed measure, built from the function \( F(x)=\left\{\begin{array}{l}0, \quad \text { when } x<0 \\ \tan ^{-1} x, \quad \text { when } 0 \leq x<1 \\ \frac{\pi}{4} x, \quad \text { when } 1 \leq x<4 \\ \pi, \quad \text { when } 4 \leq x\end{array}\right. \) 2) Find the Jordan expansion for the signed measure, built from the function \( F(x)=\left\{\begin{array}{l}0, \quad \text { when } x<0 \\ \tan ^{-1} x, \quad \text { when } 0 \leq x<1 \\ \frac{\pi}{4} x, \quad \text { when } 1 \leq x<4 \\ \pi, \quad \text { when } 4 \leq x\end{array}\right. \) What sets can be taken as \( X^{+}, X^{-} \)in Hahn decomposition??

19.6.1.2 Lebesgue measure and integration

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Problem: Let \( f: \mathbb{R} \rightarrow \mathbb{R} \). Prove that \[ \{x \in \mathbb{R}: f(x) \geq c\}=\bigcap_{n=1}^{\infty}\left\{x \in \mathbb{R}: f(x) \geq c-\frac{1}{n}\right\} . \]

19.6.1.5 Lebesgue measure and integration

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Problem: For the function \( f \), given on the segment \( [a, b] \) 1) prove that it's simple; 2) calculate the Lebesgue integral \( \int_{[a, b]} f(t) d t \), if it does exist. \[ a=0, \quad b=1, \quad f(t)=\left[\frac{1}{\sqrt{t}}\right] . \]

19.6.1.9 Lebesgue measure and integration

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Problem: By definition, calculate the Lebesgue integral \( \int_{[a, b]} f(t) d t \). \[ a=0, \quad b=3, \quad f(t)=|t-1| . \]

19.6.1.10 Lebesgue measure and integration

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Problem: Let \( f \) be an integrable function on the set \( E, g, h \) are measurable on \( E \). What can be said about the integrability of \( g, h \), if: \( f=g \cdot h, g, h>0 \).

19.6.1.11 Lebesgue measure and integration

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Problem: Calculate the Lebesgue integral \( \int_{[0,1]} f(x) d \mu \) from the function \[ f(x)=\left\{\begin{array}{c} x \cdot e^{x}, \quad x \in[0,1] \backslash Q \\ \tan ^{-1} \frac{1}{x+1}, \quad x \in Q \cap[0,1] \end{array}\right. \]

19.6.1.12 Lebesgue measure and integration

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Problem: Let \( f \) be an integrable function on \( X \). Is it true that if \( \int_{X} f(t) d \mu=0 \), then \( f(t)=0 \) almost everywhere?

19.6.1.13 Lebesgue measure and integration

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Problem: For the function \( f:[a, b] \rightarrow R \) a) find out if it's bounded; b) find the measure of the set of discontinuity points; c) find out if there is a proper or improper Riemann integral for it; d) find out if it's measurable \( f \); e) find the Lebesgue integral \( \int_{[a, b]} f(t) d t \), if it exists. \( a=-1, \quad b=1, \quad K \) is the Cantor set, \( f(t)=\left\{\begin{array}{cc}n, \quad t \in\left(\frac{1}{3^{n+1}}, \frac{1}{3^{n}}\right) \backslash K, n \in \mathbb{N}, \\ & {\left[e^{t^{2}}\right], \quad t \in K,} \\ \frac{1}{\sqrt{1+t}}, \quad & t \in\left([-1,0) \cup\left(\frac{1}{3}, 1\right)\right) \backslash K .\end{array}\right. \)

19.6.1.6 Lebesgue measure and integration

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