Dirac measure

IsThisIt???

I'm trying to find a measure that will make the measure of (-inf, 0)U (0, inf) =0.
I thought of using dirac measure so the measure at 0=1 and everywhere else is equal to 0
Is this correct or does it contradict the definition of a measure saying the measure of the null set =0

I guess another way of asking the question is 'Does the null set contain 0'

Thanks for any help

rjt

The dirac measure, defined so that any set containing zero has measure 1 and any other set to has measure 0, is a well-defined measure. It is not hard to verify the properties needed to be a measure. So there is no contradiction, and this gives you the measure you want.

Another measure that does what you want is just the zero measure: every set has measure zero. And these are basically the only possibilities (well, up to taking scalar multiples anyway).

IsThisIt??? wrote: »

I'm trying to find a measure that will make the measure of (-inf, 0)U (0, inf) =0.
I thought of using dirac measure so the measure at 0=1 and everywhere else is equal to 0
Is this correct or does it contradict the definition of a measure saying the measure of the null set =0

I guess another way of asking the question is 'Does the null set contain 0'

Thanks for any help

IsThisIt???

rjt wrote: »

The dirac measure, defined so that any set containing zero has measure 1 and any other set to has measure 0, is a well-defined measure. It is not hard to verify the properties needed to be a measure. So there is no contradiction, and this gives you the measure you want.

Another measure that does what you want is just the zero measure: every set has measure zero. And these are basically the only possibilities (well, up to taking scalar multiples anyway).

Right, was thinking it had to be the solution as we were told we can't use the zero measure as you suggested.
Thanks