# How improper Riemann and Legesgue integral associated?

I'm looking for answer to Relation between the Lebesgue integral the improper Riemann integral .
$$(L) \int_{a}^{b}f(x)dx \: \: \: \: (R)\lim_{\alpha \to a+}\int_{\alpha}^{b}f(x)dx$$ I found following theorem:
Let $$f$$ be a nonnegative continuous function. If $$f$$ is improperly Riemann integrable then it Lebesgue integrable on $$\left(a, b\right]$$ and we have $$\int_{a}^{b}f(x)dx = \lim_{\alpha \to a+}\int_{\alpha}^{b}f(x)dx$$ but there is no proof. Can somebody explain how to proof above theorem?

This question arose after reading about the Lebesgue integral of a function of arbitrary sign. Could this be somehow related to this? Also I found similar question improper Riemann integral and Lebesgue integral but its for $$\left(0, 1\right]$$

• Monotone convergence theorem? – Angina Seng May 15 at 16:15
• Take any monotonically decreasing sequence $\{\alpha_n\}$ that converges to $a$ from above and apply monotone convergence theorem. – Phoenix May 15 at 16:15

Consider the sequence $$g_n = \chi_{[\frac {1}{n},1]}.f$$ and apply the monotone convergence theorem.I,of course,assumed the domain to be $$[0,1]$$ but you get what I mean.
Edit :- You can observe why improper Riemann and Lebesgue integrals might not always be the same.The improper Riemann integral,if you note,is actually the limit of the integrals of $$\chi_{[r,b]}.f$$ as $$r\to a$$.This is actually a limit of integrals,which might not always equal the integral of the limit(Indeed,those functions converge pointwise to $$f$$ which might not necessarily be Lebesgue integrable).However,when assumptions like non negativity or integrability of $$f$$ are made,we can conclude that they are equal using the monotone and dominated convergence theorems respectively.
For $$f$$ a non-negative function on $$(a,b]$$, $$f$$ is (improperly-)Riemann integrable if \begin{align} \sup_{[x,y]} \int_x^y f(s)\mathrm{d}s < + \infty \end{align} where sur supremum is taken over all compact subinterval of $$(a,b]$$
If $$f$$ is as above, then the functions $$\mathbb{1}_{[a+\frac{1}{n},b]} \times f$$ pointwise converges to $$f$$, and increaslingly. By monotony, you can calculate the integral of the limit, and it is finite.