Category:Uniqueness of Measures

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This category contains pages concerning Uniqueness of Measures:


Let $\struct {X, \Sigma}$ be a measurable space.

Let $\GG \subseteq \powerset X$ be a generator for $\Sigma$; that is, $\Sigma = \map \sigma \GG$.

Suppose that $\GG$ satisfies the following conditions:

$(1):\quad \forall G, H \in \GG: G \cap H \in \GG$
$(2):\quad$ There exists an exhausting sequence $\sequence {G_n}_{n \mathop \in \N} \uparrow X$ in $\GG$


Let $\mu, \nu$ be measures on $\struct {X, \Sigma}$, and suppose that:

$(3):\quad \forall G \in \GG: \map \mu G = \map \nu G$
$(4):\quad \forall n \in \N: \map \mu {G_n}$ is finite


Then:

$\mu = \nu$

Pages in category "Uniqueness of Measures"

The following 3 pages are in this category, out of 3 total.