MathCS.org - Real Analysis: Example 7.4.2(e): Simple Functions

Example 7.4.2(e): Simple Functions

Are sums, differences, and products of simple functions simple?

Context

Yup - simple functions are finite sums, so they can be added, subtracted, and multiplied perfectly fine (but not divided). You don't even have to simplify the resulting functions, because we already know that simple functions can have different representations.

On the side, if A and B are two (measurable) sets then the characteristic function of the intersection of A and B is the product of the characteristic functions of A and B, i.e.

X_{A B}(x) = X_A(x) X_B(x)

Is it true that if A and B are two (measurable) sets then the characteristic function of the union of A and B is the sum of the characteristic functions of A and B?

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Example 7.4.2(e): Simple Functions