Theorem

If be increasing, right continuous function then a -algebra and a unique on such that

Note: Let be a same as , then on

, then is constant. is constant.

If is another such function then

Note: These are all called Lebesgue Stieltjes measures.

Exercise is finite on bounded intervals. Define

Show that is increasing, right continuous.

Let

Conversely, if is a Borel measure (i.e., is -additive set function on ) that is finite on all bounded intervals and define,

Misplaced &\begin{cases} & \mu((0, x]) &, x > 0 \\ & 0 &, x = 0 \\ & -\mu((0, -x]) &, x < 0 \end{cases}$$ Then $F_{\mu}$ is increasing, right continuous and the Lebesgue Stieltjes measure corresponding to $F_{\mu}$ on $\mu$. **Exercise** Show that right continuity of $F$ is necessary. **Exercise** $$F(x) = \begin{cases} 1 &, x \geqslant 0 \\ 0 &, x < 0 \end{cases}$$ Find the corresponding $\sigma$-algebra $\mathcal{M}_F$ & the measure $\mu_F$. ____ # Definition (Complete) Let $\mathcal{F} \subseteq \mathcal{P}(\Omega)$ be a $\sigma$-algebra & $\mu : \mathcal{F} \to [0, \infty]$. The $\sigma$-algebra is called **complete** wrt $\mu$ (the measure space $(\Omega, \mathcal{F}, \mu)$ is complete) if for all $F \subseteq A$, $A \in \mathcal{F}$ & $\mu(A) = 0$ implies $F \in \mathcal{F}$. Define $$\mathcal{F}_{\mu} = \{ A \cup N : A \in \mathcal{F}, N \subseteq E \in \mathcal{F}, \mu(E) = 0 \}$$ **Exercise** Prove $\mathcal{F}_{\mu}$ is a $\sigma$-algebra. Define $\bar{\mu}$ by $\bar{\mu} : \mathcal{F}_{\mu} \to [0, \infty]$ $$\bar{\mu}(A \cup N) = \mu(A)$$ **Exercise** Prove $\bar{\mu}$ ___ **Note:** $A \cup N = B \cup M$ $$\Rightarrow \mu(A) = \mu(B)$$ $A, B \in \mathcal{F}, M \subseteq E, \mu(E) = 0$ $N \subseteq F, \mu(F) = 0$ ____ **Exercise** :- Prove $(\Omega, \mathcal{F}_{\mu}, \bar{\mu})$ is a complete measure space.

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