Determine the Lebesgue measure of a set of numbers from the range [0,1] for which there are such decimal increments of $0,1,2,3,\ldots,$ that each of them appears after the decimal point at least once.
2026-03-24 23:42:43.1774395763
Lebesgue measure on interval
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1
Having no idea what a "decimal increment" is, I find the statement unclear. I think you mean this: Find the measure of the set of numbers $x\in[0,1]$ such that each digit $0,\dots,9$ appears at least once in the decimal expansion of $x$.
If that's the question the answer is $1$.
Hint: It's enough to show that $m(S)=0$, if $S$ is the set of $x$ such that $0$ does not appear in the decimal expansion of $x$. (Then the same holds for any other digit in place of $0$, and the union of ten null sets is a null set...)
Of course a number can have more than one decimal expansion, so the problem is really not well defined. But there are only countably many numbers with more than one expansion (proof?), so we can ignore them.
Ignoring numbers with more than one expansion, the set of numbers that do not have $0$ as the first digit is simply $[1/10,1]$. Now figure out what the set of numbers without a zero in the first two digits is...