Solving Lebesgue Integral Homework with Finite Support

[sbashrawi]

Homework Statement

Let the function f be nonnegative and integrable over E and
$$\epsilon$$ > 0. Show there is a simple function $$\varphi$$ on E that has finite support,
0< $$\varphi$$< f on E and

integration |f − $$\varphi$$|<$$\epsilon$$

Homework Equations

The Attempt at a Solution

 

[lippyka]

Let \varphi(x) = \sum_{i=1}^{n} c_i \cdot \chi_{A_i}(x) where A_i is a measurable subset of E and c_i are constants. Then, we can choose c_i's and A_i's such that 0<c_i<f(x) on E and \sum_{i=1}^{n} c_i \cdot \mu(A_i) <\epsilonThis condition is sufficient for the given equation to be satisfied.