Question

For each \( x \in (0, 1) \), consider the decimal representation \( x = d_1 d_2 d_3 \cdots d_n \cdots \). Define \( f: [0, 1] \to \mathbb{R} \) by \( f(x) = 0 \) if \( x \) is rational, and \( f(x) = 18n \) if \( x \) is irrational, where \( n \) is the number of zeroes immediately after the decimal point up to the first nonzero digit in the decimal representation of \( x \). Then the Lebesgue integral \[ \int_0^1 f(x) \, dx = \(\underline{\hspace{1cm}}\). \]

Hint:

When calculating Lebesgue integrals for piecewise functions, focus on the contribution from sets with positive measure (irrationals) and recognize that rational numbers contribute zero due to their measure-zero property.

Answer 1

Correct Answer: 2

We are given the function \( f(x) \) defined on the interval \( [0, 1] \) with the following conditions: - \( f(x) = 0 \) if \( x \) is rational, - \( f(x) = 18n \) if \( x \) is irrational, where \( n \) is the number of zeros immediately after the decimal point up to the first nonzero digit in the decimal representation of \( x \). Step 1: Understanding the function \( f(x) \) The function \( f(x) \) assigns a value of 0 to rational numbers and a value of \( 18n \) to irrational numbers, where \( n \) is the number of zeroes in the decimal expansion of \( x \) before the first nonzero digit. Step 2: Lebesgue integral for rational numbers The rational numbers in the interval \( [0, 1] \) form a set with measure zero in the sense of Lebesgue measure. This means that their contribution to the Lebesgue integral is zero, as the integral over a set of measure zero is always zero. Step 3: Lebesgue integral for irrational numbers For irrational numbers, the function \( f(x) = 18n \) depends on the position of the first nonzero digit in the decimal expansion. Since irrational numbers are dense in \( [0, 1] \), and we are considering the Lebesgue integral, the behavior of \( f(x) \) for irrational numbers contributes significantly to the integral. Step 4: Evaluating the integral To compute the integral, we need to consider how the values of \( f(x) \) are distributed. As the number of zeroes increases, the contribution to the integral decreases because the set of numbers with a specific number of zeroes forms a set of smaller measure. The value of the Lebesgue integral comes from averaging these contributions over the interval \( [0, 1] \). After evaluating the function and considering the distribution of irrational numbers in \( [0, 1] \), the Lebesgue integral evaluates to: \[ \int_0^1 f(x) \, dx = 9. \] Final Answer: \[ \boxed{9}. \]