So, while solving problems on double integral I came across this weird problem:

Let $$f(x,y) = 1/2,\ \forall x\in \mathbb Q$$ $$f(x,y) = \ y, \ \forall x\in \mathbb Q^c$$ then find $$\int_{0}^1\left(\int_0^{1}fdx\right)dy$$

I have no idea how to approach this problem, First of all I am not sure whether this integral even exist or not .

Can anyone please explain to me how to think in this question ?

Thank you.


You need Lebesgue Integration to be able to do this, as the Riemann integral does not converge.

Let $1_{\mathbb{Q}^c}(x)$ be the indicator function of the irrationals. From $$\int_0^1 c \cdot 1_{\mathbb{Q}^c}(x)\,dx = c, \qquad \text{and} \qquad \int_0^1 c \cdot 1_{\mathbb{Q}}(x)\,dx = 0 ,$$ it follows (since $y$ is constant in the inner integral) that your integral is $$\int_0^1 y\,dy= \frac{1}{2}.$$

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    $\begingroup$ You mean $\mathbb Q^{\mathrm c}$; actually $\Bbb Q^{\Bbb C} = \{\Bbb C\to\Bbb Q\}$, the set of all functions from $\Bbb C$ to $\Bbb Q$. $\endgroup$ – gen-ℤ ready to perish Aug 21 '19 at 20:38
  • $\begingroup$ @gen-zreadytoperish Is that what I originally wrote! Ha! $\endgroup$ – Dzoooks Aug 21 '19 at 22:10
  • $\begingroup$ Ain’t no thing but a chicken wing $\endgroup$ – gen-ℤ ready to perish Aug 22 '19 at 0:11

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