My questions is if $X$ is measurable space and we define two measures on X. We wan to take $max$ of them.Is it measure I have falling that it is not correct ,however ,one who has an idea that it will be help.
Max between two measures
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real-analysis
1 Answers
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Indeed. Consider the set $A = \{0,1\}$, with $\Sigma$-algebra $\mathcal P(A)$ and measures $\mu_0,\mu_1$ defined by $\mu_0(\{0\}) = \mu_1(\{1\}) = 1$. Then if we define $\mu = \max\{\mu_0,\mu_1\}$, we would have $$ \mu(A) = \mu(\{0\}) = \mu(\{1\}) = 1, $$ contradicting countable (even finite) additivity of the measure.
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0For clarity: one completes the definition of the measures by setting $\mu_0(\{1\}) = \mu_1(\{0\}) = 0$. It follows that $\mu (A) = \max \{ \mu_0 (A) , \mu_1 (A)\} = \max \{1, 1\} = 1$, but also $\mu (A) = \mu (\{0\}) + \mu(\{1\}) = \max \{ \mu_0 (\{0\}) , \mu_1 (\{0\})\} + \max \{ \mu_0 (\{1\}) , \mu_1 (\{1\})\} = 1 + 1 = 2$. – 2018-01-27