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 .