Suppose I have two positive Borel measures on [ 0 , ∞ ], say μ and ν and I know that ν ( [ 0 , x ] ) = 0 implies μ ( [ 0 , x ] ) = 0 for any x and that ν ( ( a , b ] ) = 0 implies μ ( ( a , b ] ) = 0 for any pair a , b.Does this imply that μ &lt;&lt; ν? I would think so but I have trouble proving it, I thought that the monotone class theorem was the answer but I don't see how the sets with the property ν ( E ) = 0 ⟹ μ ( E ) = 0 are closed under monotone intersections.We may assume that the measures are finite, if that helps anything.

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Suppose I have two positive Borel measures on   [  0  ,  ∞  ], say   μ and   ν and I know that   ν  (  [  0  ,  x  ]  )  =  0 implies   μ  (  [  0  ,  x  ]  )  =  0 for any   x and that   ν  (  (  a  ,  b  ]  )  =  0 implies   μ  (  (  a  ,  b  ]  )  =  0 for any pair   a  ,  b.Does this imply that   μ  &amp;lt;&amp;lt;  ν? I would think so but I have trouble proving it, I thought that the monotone class theorem was the answer but I don&#039;t see how the sets with the property   ν  (  E  )  =  0  ⟹  μ  (  E  )  =  0 are closed under monotone intersections.We may assume that the measures are finite, if that helps anything.

scipionhi · Accepted Answer

The circumstance is worse than what I said. Consider any action&amp;nbsp;ν&amp;nbsp;on a nondegenerate interval, that is never 0. (the Lebesgue measure works, as do many finite measures). If&amp;nbsp;ν&amp;nbsp;has sets of measure 0, then take one such set&amp;nbsp;E, let&amp;nbsp;λ&amp;nbsp;be a probability measure on&amp;nbsp;E, and let&amp;nbsp;μ&amp;nbsp;=&amp;nbsp;ν&amp;nbsp;+&amp;nbsp;λ. Then&amp;nbsp;μ&amp;nbsp;and&amp;nbsp;ν&amp;nbsp;satisfy the condition on intervals, but&amp;nbsp;μ&amp;nbsp;is not absolutely continuous with respect to&amp;nbsp;ν.

Suppose I have two positive Borel measures on [ 0 , <mi mathvariant="normal">∞<!

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