Compute Lebesgue-Stieltjes integral.

Question

I am a bit confused, because I've never seen these notations.

Task to compute following integral: $$\int_{-\infty}^{+\infty} |f(x)|\,d\nu(x), $$ where $$ \nu(x) = \begin{cases} -2& \text{, with weight 0.2; }\\ 0& \text{, with weight 0.3;}\\ 1& \text{, with weight 0.5. } \end{cases} $$

I am not familiar with this... Can somebody explain how measure can have some probability function in it? And how should I compute this integral? Let $f(x) = x$, for example.

Answer 1

Let $v_i(x)$ be the function with weight $w_i$, then $$\int_{-\infty}^{+\infty} |f|dv=\sum_{i=i}^n w_i|f(v_i(x))|.$$ In this case $$\int_{-\infty}^{+\infty} |f|dv=0.2|f(-2)|+0.3|f(0)|+0.5|f(1)|.$$

Rember that a probability space $(\mathcal{P},\mu )$ is a measure space such that $\mu:P(E) \to [0,1]$ (where $P(E)$ is the sets of all subsets of the universe set $E$) and $$\mu \left( \bigcup_{i \in I} E_i \right)=\sum_{i \in I} \mu(E_i).$$

Answer 2

Noting that the sum of measures defined over the same sigma algebra is also a measure on the same sigma algebra, you can think of $\nu(A)$ as being $$ \nu(A) = 0.2 \times \delta_{-2}(A) + 0.3 \times \delta_0(A) +0.5 \times \delta_1 $$ where $\delta_x$ denotes the Dirac measure concentrated at point $x$ and $A$ is a measurable set. There's nothing exotic about it at all!