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I'm having some trouble with this problem.

Let $$f(x)= \begin{cases} 1 &\text{for}\,\, x = \frac{1}{n}\,,\, n=1,2,\cdots \\ 2 &\text{otherwise} \end{cases}$$ Compute the value of the Lebesgue integral $$\int_0^5 f(x)dm(x)$$

Clearly, $f$ is a simple function. In this case, I know that $$\int_0^5 f(x)dm(x) = \sum_{i=1}^2 \alpha_i m^*(A_i)$$ where $m^*$ is the Lebesgue outer measure and $A_1, A_2$ correspond to the Lebesgue measurable sets.

I know that $\alpha_1 = 1$ and $\alpha_2 = 2$, but I'm not sure how to go about computing the measures of these sets. Any help would be greatly appreciated. Thanks.

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  • $\begingroup$ Maybe use $f$ may as well have value $2$ everywhere? $\endgroup$ May 2, 2015 at 14:36

1 Answer 1

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the L-measure of (0,5) is 5

The L-measure of a single point is 0, and such is the L-measure of any enumerable set of points

hence the L-measure of $A_1=\{1,2,3,4\}$ (or {1,...,5}) is 0

the L-measure of its complement is then 5, and the integral is 10

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