Integration of function regarding decimal representation

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Let $x=0.n_1n_2...$ be the decimal representation of $xin[0,1]$. Compute with justification $int_0^1f(x)dx$, where a) $f(x)=max_iinmathbbN|n_i-n_i+1|$ and b) $f(x)=max_iinmathbbNsum_k=i^inftydfracn_k2^k-i$.



I can only show that the measure of $x$ with at least one 9 in its decimal representation is 1. I have no idea about the difference, let alone the b) part.




real-analysis lebesgue-integral lebesgue-measure




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  • Use the same proof in base $100$ to show that the measure of $x$ with at least one $90_100$ is $1$.
    – Yves Daoust
    Aug 13 at 7:28














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Let $x=0.n_1n_2...$ be the decimal representation of $xin[0,1]$. Compute with justification $int_0^1f(x)dx$, where a) $f(x)=max_iinmathbbN|n_i-n_i+1|$ and b) $f(x)=max_iinmathbbNsum_k=i^inftydfracn_k2^k-i$.



I can only show that the measure of $x$ with at least one 9 in its decimal representation is 1. I have no idea about the difference, let alone the b) part.




real-analysis lebesgue-integral lebesgue-measure




share|cite|improve this question




















  • Use the same proof in base $100$ to show that the measure of $x$ with at least one $90_100$ is $1$.
    – Yves Daoust
    Aug 13 at 7:28












up vote
0
down vote

favorite









up vote
0
down vote

favorite











Let $x=0.n_1n_2...$ be the decimal representation of $xin[0,1]$. Compute with justification $int_0^1f(x)dx$, where a) $f(x)=max_iinmathbbN|n_i-n_i+1|$ and b) $f(x)=max_iinmathbbNsum_k=i^inftydfracn_k2^k-i$.



I can only show that the measure of $x$ with at least one 9 in its decimal representation is 1. I have no idea about the difference, let alone the b) part.




real-analysis lebesgue-integral lebesgue-measure




share|cite|improve this question












Let $x=0.n_1n_2...$ be the decimal representation of $xin[0,1]$. Compute with justification $int_0^1f(x)dx$, where a) $f(x)=max_iinmathbbN|n_i-n_i+1|$ and b) $f(x)=max_iinmathbbNsum_k=i^inftydfracn_k2^k-i$.



I can only show that the measure of $x$ with at least one 9 in its decimal representation is 1. I have no idea about the difference, let alone the b) part.





real-analysis lebesgue-integral lebesgue-measure





share|cite|improve this question











share|cite|improve this question




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asked Aug 13 at 7:10









Leonardo

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  • Use the same proof in base $100$ to show that the measure of $x$ with at least one $90_100$ is $1$.
    – Yves Daoust
    Aug 13 at 7:28
















  • Use the same proof in base $100$ to show that the measure of $x$ with at least one $90_100$ is $1$.
    – Yves Daoust
    Aug 13 at 7:28















Use the same proof in base $100$ to show that the measure of $x$ with at least one $90_100$ is $1$.
– Yves Daoust
Aug 13 at 7:28




Use the same proof in base $100$ to show that the measure of $x$ with at least one $90_100$ is $1$.
– Yves Daoust
Aug 13 at 7:28















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