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**JJMC89** Denote by $\displaystyle m$ the Lebesgue measure on $\displaystyle X = (0, 1)$.

$\displaystyle d(\cdot)$ is the Dirichlet function.

Consider the sequence of functions on $\displaystyle (0, 1)$ defined as follows: $\displaystyle f_N(x)=\left\{\begin{array}{cc}0,&\mbox{ if }

x\mbox{ is irrational or }x = \frac{k}{n}\mbox{ (in lowest terms) and }n>N\\1, & \mbox{ if }x = \frac{k}{n}\mbox{ (in lowest terms) and }n \leq N\end{array}\right.$

1) Prove that $\displaystyle f_N(x) < f_{N+1}(x), \ x \in (0, 1)$, and that $\displaystyle \lim_{N\rightarrow \infty} f_N(x) = d(x), \ x \in

(0, 1)$.

2) Are the functions $\displaystyle f_N$ Riemann integrable? If your answer is yes, what is $\displaystyle \int_0^1 f_N (x)dx$?

3) Are the functions $\displaystyle f_N$ Lebesgue integrable? If your answer is yes, what is the Lebesgue integral $\displaystyle \int f_N(x)dm$?

4) Is it true that $\displaystyle \lim \int_0^1 f_N(x)dx = \int_0^1 d(x)dx$ (the Riemann integral)?

5). Is it true that $\displaystyle \lim \int_0^1 f_N(x)dx \leq \int d(x)dm$ (the Lebesgue integral)?