# Thread: Continuous function => Measurable

1. ## Continuous function => Measurable

If a function f:[a,b]->R is continuous a.e, then show that f is a measurable function.

2. Originally Posted by Chandru1
If a function f:[a,b]->R is continuous a.e, then show that f is a measurable function.
The definition of continuity is that $\displaystyle f^{-1}$ takes open sets to open sets. f is a measurable function if $\displaystyle f^{-1}$ takes open sets to measurable sets. But, every open set is measurable.

Thus, all that remains is to show that the a.e. condition doesn't effect measurability.

Define g:[a,b]-->R to be:

g(a) = f(a) if f is continuous at a
g(a) = $\displaystyle \lim_{x->a}{f(x)}$ if f is not continuous at a

Then g is a continuous function, so g is measurable. But g = f a.e. thus, f is also measurable.

Note: You may have to show g is well-defined

3. Originally Posted by southprkfan1
The definition of continuity is that $\displaystyle f^{-1}$ takes open sets to open sets. f is a measurable function if $\displaystyle f^{-1}$ takes open sets to measurable sets. But, every open set is measurable.
Isn't a function measurable iff the preimage of measurable sets is measurable?