Riemann integrable vs Lebesgue integrable

The proof for Riemann could be easier.

Since $\int_a^bf(x)dx>0$, there is a Partition $P$ such that $L(P,f)>0$, where $L(P,f)$ means Darboux lower sum. So there must be a interval, where $\inf f>0$


The theorem is false in the Lebesgue setting. Consider $$f(x) = \begin{cases} 0 & x \in \mathbb{Q} \\ 1 & \textrm{otherwise} \end{cases} .$$ Your very first sentence of your "proof" is wrong.

EDIT: I just realized this was meant to be a proof for Riemann-integrable functions, not Lebesgue.


For Lebesgue integrals, this is not even true up to modification on a set of measure zero. Take for instance the indicator function of a "fat" Cantor set (cf. page 39 of Folland's 'Real Analysis').

Tags:

Real Analysis

Measure Theory

Related

If a series $\sum\lambda_n$ of positive terms is convergent, does the sequence $n\lambda_n$ converge to $0$? To prove $n\binom{2n-1}{n-1}$ is divisible by $n(2n-1)$ Applying law of total probability to conditional probability Additive function $T: \mathbb{R} \rightarrow \mathbb{R}$ that is not linear. Is it always possible to create a intuition for abstract algebra theorems? $\int_{0}^{\frac{\pi}{2}}\frac{d\theta}{a + \sin^2(\theta)} = \frac{\pi}{2\sqrt{a(a+1)}}$, for $a > 0$ What is the significance of the quotient $m/m^2$? Wilson's theorem Prime Generator Distribution of $X\sqrt {2Y}$ where $X\sim N (0,1)$ and $Y\sim \operatorname{Exp} (1)$ Finding the general Taylor Series of a function Must a Set's size be a natural number, and would the contrary invalidate a solution? In a field why does the multiplicative identity have an additive inverse, whereas the additive identity doesn't have a multiplicative inverse?

Recent Posts