Define the distribution function $\Phi_{f}$ of a non-negative measurable function $f$ on the interval $I=[0,1]$. Show that $\Phi_{f}$ is a decreasing non-negative function on $[0, \infty]$ which is continuous on the right.

Define the Lebesgue integral $\int_{I} f d m$. Show that $\int_{I} f d m=0$ if and only if $f=0$ almost everywhere.

Suppose that $f$ is a non-negative Riemann integrable function on $[0,1]$. Show that there are an increasing sequence $\left(g_{n}\right)$ and a decreasing sequence $\left(h_{n}\right)$ of non-negative step functions with $g_{n} \leqslant f \leqslant h_{n}$ such that $\int_{0}^{1}\left(h_{n}(x)-g_{n}(x)\right) d x \rightarrow 0$.

Show that the functions $g=\lim _{n} g_{n}$ and $h=\lim _{n} h_{n}$ are equal almost everywhere, that $f$ is measurable and that the Lebesgue integral $\int_{I} f d m$ is equal to the Riemann integral $\int_{0}^{1} f(x) d x$.

Suppose that $j$ is a Riemann integrable function on $[0,1]$ and that $j(x)>0$ for all $x$. Show that $\int_{0}^{1} j(x) d x>0$.