Suppose that X is a real-valued random variable on the probability space ( Ω , F , P ) with a cumulative distribution function F X ( x ) = P [ X ≤ x ]. Can we conclude from some measure theoretic property that P [ X ≤ x ] = P [ X &lt; x ]? The measure zero of singleton points is certainly true for many well-known measures, but can we conclude that in general, subtracting a finite number of points from ( − ∞ , x ] does not change the probability of ( − ∞ , x ]?I'm asking this because my reading material hasn't explicitly taken care of this, and from other courses I know that P [ − x ≤ X ≤ x ] = F X ( x ) − F X ( − x ), but arguing only with the general properties of probability measures I know of, yields P [ − x ≤ X ≤ x ] = P [ X ≤ x ] − P [ X &lt; − x ], where the RHS would simplify to include only the CDF of X, if the finite difference of points doesn't matter.

Question

Suppose that X is a real-valued random variable on the probability space   (  Ω  ,      F    ,      P    ) with a cumulative distribution function       F    X    (  x  )  =      P    [  X  ≤  x  ]. Can we conclude from some measure theoretic property that       P    [  X  ≤  x  ]  =      P    [  X  &amp;lt;  x  ]? The measure zero of singleton points is certainly true for many well-known measures, but can we conclude that in general, subtracting a finite number of points from   (  −  ∞  ,  x  ] does not change the probability of   (  −  ∞  ,  x  ]?I&#039;m asking this because my reading material hasn&#039;t explicitly taken care of this, and from other courses I know that       P    [  −  x  ≤  X  ≤  x  ]  =      F    X    (  x  )  −      F    X    (  −  x  ), but arguing only with the general properties of probability measures I know of, yields       P    [  −  x  ≤  X  ≤  x  ]  =      P    [  X  ≤  x  ]  −      P    [  X  &amp;lt;  −  x  ], where the RHS would simplify to include only the CDF of X, if the finite difference of points doesn&#039;t matter.

Kristina Petty · Accepted Answer

You need the measure to be continuous so it has no atoms. It’s not a general feature of all measures.

Suppose that X is a real-valued random variable on the probability space ( <mi mathvariant="n

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