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

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- Apr 10th 2010, 01:39 AMChandru1Continuous function => Measurable
If a function f:[a,b]->R is continuous a.e, then show that f is a measurable function.

- Apr 10th 2010, 08:22 AMsouthprkfan1
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 - Apr 10th 2010, 08:28 AMmaddas