Let be Lebesgue integrable functions on the real line R and let , .
Show that is measurable on R.
i dont even know how to start. help is appreciated so much.
If f and g are integrable then (by definition) they must be measurable.
Start by showing that the function is measurable (for fixed x). You should be able to do that from the definition of measurability. You then need to show that the product of that function with g is measurable. To show that the product of two measurable functions is measurable, first show (from the definition) that if is measurable then so is . Then, if and are measurable, so is .
If E is a measurable subset of , then is a measurable subset of . Reason: it is equal to the set (which is measurable) rotated through an angle , and rotations preserve measurability. Therefore if f is measurable, so is the function . The function is clearly measurable. Hence so is the product .
The reason for the 45 degrees is that the function is obviously constant along lines of the form const. So if you rotate the set through 45 degrees then you get a set that is constant along lines parallel to one of the axes. (The factor comes in because it is the cosine of 45 degrees.)