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I was given I tricky problem where I am really confused with the possible solution.

let $\mu$ be a measure defined on $[-1,2]$ as follow: on the set $[-1,2]/\{-1,0,1,2\}, \mu(A) = \int_{A}x^2 dx.$ On the set $B = \{-1,0,1,2\}, \mu(x) = \frac{1}{4}, \forall x\in B$.

The question is asking me to find $\int_{[-1,2]}xd\mu(x)$.

It is pretty clear to me if $\mu(x)$ is just an ordinal Lebesgue measure. However, with this non-standard form of measure, I am not sure how to start. Is this some form of Lebesgue–Stieltjes integration?

Please help! Thank you so very much!

Brain Lim
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  • Hey for anyone who is interested in this question, I found this very very helpful: https://math.stackexchange.com/questions/32217/how-to-compute-riemann-stieltjes-lebesgue-stieltjes-integral?rq=1 – Brain Lim Nov 07 '20 at 00:54

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$\mu =\mu_1+\frac 1 4 \delta_{-1}+\frac 1 4 \delta_{0}+\frac 1 4 \delta_{1}+\frac 1 4 \delta_{2}$ where $\delta_x$ denotes the measure defined by $\delta_x(E)=1$ or $0$ according as $x\in E$ or not and $\mu_1(E)=\int_E x^{2}dx$. For any $\mu -$ intergrable function $f$ we have $\int fd\mu=\int x^{2}f(x)dx+\frac 1 4 f(-1)+\frac 1 4 f(0)+\frac 1 4 f(1)+\frac 1 4 f(2)$.

Kavi Rama Murthy
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  • Thx Kavi, could you further explain how do I handle the situation on non-Lebesgue measure Lebesgue integrals? Much appreciated! – Brain Lim Nov 07 '20 at 00:06