I have recently found a problem in measure and integration which is given below :---
Let $f:[0,1]\to \Bbb R$ be monotone increasing function with $f(0)=0$ and $f(1)=1$. Suppose, $\mu$ denotes the Borel measure on $[0,1]$ such that, for all $0\leq a<b\leq 1$ we have $$\mu\big((a,b]\big)=\text{ Cardinality of } \bigg\{ x\in [0,1]:a<\lim_{h\to 0+}\frac{f(x+h)-f(x)}{h}\leq b\bigg\}.$$ Prove that, $$\int_{[0,1]}t^pd\mu<\infty,\text{ for all } p>1.$$ I don't know how do I start with this problem. Any help will be appreciated.
This is not true at all. Let $f(x)=\frac {e^{x}-1} {e-1}$. Then $\mu (a,b])$ is the cardinality of $\{x \in [0,1]: a(e-1) <e^{x} <b(e-1)\}$ which is $\infty$ whenever $a <b$ and $\frac 1 {e-1} <b<1$. This makes $\int t^{p}d\mu \geq \int_{1/(e-1)}^b t^{p}d\mu =\infty$