Lebesgue and Riemann integrals two proofs 1.
Let X be a finite closed interval [a,b] in R, let X be the collection of Borel sets in X and let λ be a Lebesgue measure on X.
If f is a nonnegative function on X, show that
∫ fdu =∫a->b f(x)dx
I know that the right side denotes the Riemann integral of the function. 
and I am given the hint that first establish this equality for a nonnegative step function (a linear combo of characteristic function of intervals.)
2.
A slight change, let X=[0, +∞) let X be the Borel subsets of X let λ be a Lebesgue measure on X.
If f is a nonnegative function on X, show that 
∫ fdλ =lim b ->+∞ ∫0->b f(x)dx
It means that the Lebesgue and the improper Riemann integrals coincide. But how to show 
∫ fdλ =lim b ->+∞ ∫0->b f(x)dx
Please help prove the two similar questions in details.
Thanks!
 A: (1) If $f : X= [c,d] \to \mathbb{R}$ is a nonnegative Riemann integrable function, then
$$\int_c^d f(x) \, dx  = \sup_{\phi  \leqslant f} \int_c^d\phi(x) \, dx = \inf_{\psi  \geqslant f} \int_c^d\psi(x) \, dx ,$$
where $\phi$ and $\psi$ are step functions.  This is straightforward to show using upper and lower Darboux sums and integrals.
Since any step function is a simple function, we have for simple functions $\hat{\phi}$ and $\hat{\psi}$
$$\sup_{\phi  \leqslant f} \int_c^d\phi(x) \, dx \leqslant \sup_{\hat{\phi}  \leqslant f} \int_{[c,d]}\hat{\phi} \leqslant \inf_{\hat{\psi}  \geqslant f} \int_{[c,d]}\hat{\psi}  \leqslant \inf_{\psi  \geqslant f} \int_c^d\psi(x) \, dx ,$$
and Riemann integrability implies Lebesgue integrability with
$$\int_{[c,d]}f= \int_c^df(x) \, dx .$$
The converse is not true.  Lebesgue integrability does not imply Riemann integrability -- even on a finite interval. (Consider the Dirichlet function).
(2)  Note that by the monotone convergence theorem, if $f$ is nonnegative and is improperly Riemann integrable, then
$$\int_{[a,\infty)}f = \lim_{n \to \infty} \int_{[a,\infty)} f 1_{[0,n]} = \lim_{n \to \infty} \int_a^n f(x) \, dx = \int_a^{\infty} f(x) \, dx$$
since the Riemann and Lebesgue integrals coincide on bounded intervals.
