Dirac delta integral on the bounded set

Question

How to solve that an integral? $$\int_{[0,10)}x^2 d\mu $$ where $$\mu = \sum_{k=1}^{4} {{1}\over{k}} \delta_{k}$$ I know that $$\int_{X} {f(x) \delta_{x_0} (dx)}= f(x_{0}) $$ but in that example I don't know how to do it on the interval (0,10]. Does anyone know how to solve this?

Answer 1

Most likely how you're intended to solve this is to know the following two rules, or at least for them to be "intuitively obvious" to you if you haven't covered them in class specifically:

1. $\int fd\delta_x=f(x)$
2. $\int fd(\sum\lambda_i\mu_i)=\sum\lambda_i\int fd\mu_i$

You can use these rules to work out your integral.