Equivalent statements related to Lebesgue measurability
Let (X,@) be a measurable space.
Statement 1:
it is given that for each a in R the set
{x in X | f(x) <= a } is measurable
Statement 2: f is measurable
We need to prove that statement 1 implies 2.
My approach
Since for each a in R the set
{x in X | f(x) <= a } is measurable = intersection of of n from 1 to infinity
complement of {x in X | f(x) > a } is measurable
Since
{x in X | f(x) > a } is measurable, its complement is also measurable
==> f is measurable
Is this a correct approach?
Many thanks for your feedback
Statement 1:
it is given that for each a in R the set
{x in X | f(x) <= a } is measurable
Statement 2: f is measurable
We need to prove that statement 1 implies 2.
My approach
Since for each a in R the set
{x in X | f(x) <= a } is measurable = intersection of of n from 1 to infinity
complement of {x in X | f(x) > a } is measurable
Since
{x in X | f(x) > a } is measurable, its complement is also measurable
==> f is measurable
Is this a correct approach?
Many thanks for your feedback
(edited 10 months ago)
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