A simple function is a measurable function whose image is a finite set. Its value sets form a finite measurable partition. For nonnegative , its Lebesgue integral is , with the convention . The pieces need not be intervals, or even connected sets. See MIT lecture 10, Definitions 100 and 106.
For example, on let be on , on , and on . Its Lebesgue integral is . Its superlevel sets have Lebesgue measure for , for , and thereafter. The layer cake representation gives the same answer: . These two calculations are the building blocks of the general Lebesgue integral.
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In mathematics, a simple function is typically defined as a function that can be expressed as a finite sum of simple components. The most common context where "simple function" is used is in measure theory, where a simple function is a measurable function that takes only a finite number of values. ### Characteristics of Simple Functions: 1. **Finite Range**: A simple function only assumes a finite set of values. For instance, the function can take values \( c_1, c_2, ...