Question

Let \( f : [0, 1] \to {R} \) be a function. Which one of the following is a sufficient condition for \( f \) to be Lebesgue measurable?

• A. \( |f| \) is a Lebesgue measurable function
• B. There exist continuous functions \( g, h : [0, 1] \to {R} \) such that \( g \leq f \leq h \) on \( [0, 1] \)
• C. \( f \) is continuous almost everywhere on \( [0, 1] \)
• D. For each \( c \in {R} \), the set \( \{ x \in [0, 1] : f(x) = c \} \) is Lebesgue measurable

Hint:

For Lebesgue measurability, consider whether the function is continuous except on a measure-zero set.

Answer 1

The Correct Option is C

Step 1: Sufficient conditions for measurability. A function \( f \) is Lebesgue measurable if it is continuous almost everywhere because the set of discontinuities has measure zero. Step 2: Analyzing options. Option (3) is sufficient because continuity almost everywhere ensures that \( f \) is measurable. Step 3: Conclusion. The sufficient condition is \( {(3)} \).