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Let $f:(0,\infty)\rightarrow [0,\infty)$, $f(x)=\left\{\begin{array}{ll} |\sin(x)| & \text{ if $x\in \mathbb Q$}\\ 4x & \text{ if $x\in[0,1]\backslash \mathbb Q$}\\ 0 & \text{ if $x\in (1,\infty)\backslash \mathbb Q$} \end{array}\right.$

I want to determine $\int_{(0,\infty)}fd\lambda$. Is it possible to calculate $\int_\mathbb Q |\sin(x)|d\lambda+\int_{[0,1]\backslash\mathbb Q}4xd\lambda+\int_{(1,\infty)\backslash\mathbb Q}0 d\lambda$? The fist integral is $0$ because $\mathbb Q$ is a null set, the second integral is $2$ and the third integral is $0$. So $\int_{(0,\infty)}fd\mu=2$. Is this correct?

Stephen Donovan
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Robert
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1 Answers1

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Yes.

You can present your calculation in the simpler way. Consider the function $g(x)=4x\cdot 1_{[0,1]}(x)$. Then $f=g$ $\lambda$-a.e. and $$ \int f\;d\lambda = \int g\;d\lambda =\int_0^14x\;dx = 2 $$