Is it even integrable?

Relevant wiki: Lebesgue Integration

$S$ is defined as the set of rational numbers. Our function is equal to $x$ whenever $x$ is an irrational number, and $2$ when it isn't. The set $S$ has a measure of zero over the real numbers ( $\mu{(S)} = 0$ ). In other words, the amount of rational numbers is negligible in comparison to that of the real numbers, and thus $|\mathbb{R}| - |S|$ is essentially equal to $|\mathbb{R}|$ . Almost all real numbers are irrational.

Thus, we say that the function is defined as $f(x) = x$ almost everywhere in the domain of $f$ and $\mu{ ( \{ x : x \in S \} ) } = \mu{ ( \{ x : f(x) = 2 \} ) } = \mu{ ( \{ x : x \neq f(x) \} ) } = 0$ . We can thus rewrite the integral:

$\displaystyle \int_{\mathbb{R^+ \leq 10}}{f(x) \ dx} = \int_{\mathbb{R^+ \leq 10}} {x \ dx} = \dfrac{x^2}{2} \bigg\rvert_0^{10}.$

Hence, the Lebesgue integral evaluates to $50._{\blacksquare}$

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Note:

• Here, $|X|$ refers to the cardinality of a set $X$ (the number of elements in $X$ ).

• $\mu{(X)}$ refers to the measure of a set $X$ over some interval.

• Let $\mathbb{I}$ denote the set of irrational numbers. Then $|\mathbb{R}| - |S| = |\mathbb{I}|$ .