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A number $x\in\mathbb{R}$ is said to be algebraic if it is the root of a polynomial over $\mathbb{Q}$. Define $A:[0,1]\rightarrow \{0,1\}$ as $$A(x)=1 \text { if } x \text{ is algebraic, } A(x)=0\text{ if } x \text{ is not algebraic, }$$

What is the value of $\int_{0}^{1} A(x)dx$ ?

  • 2
    The set of algebraic numbers is countable... – 2017-01-20
  • 1
    It depends on whether you are making a Riemann integral or a Lebesgue integral. – 2017-01-20

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The set of all algebraic numbers is countable .use it to deduce the value=0