## 7.4. Lebesgue Integral

### Example 7.4.4(b): Lebesgue Integral for Simple Functions

Find the Lebesgue integral of a step function, i.e. a function

A step function is a simple function, i.e.
*s*such that*s(x) = c*for_{j}*x*and the_{j-1}< x < x_{j}*{ x*form a partition of_{j}}*[a, b]*.wheres(x) = c_{j}X_{j}(x)

*X*is the characteristic function of the set

_{j}*(x*. All sets are measurable with finite measure so that by definition:

_{j-1}, x_{j})which is again the same answer as for the Riemann integral.s(x) dx = c_{j}m(X_{j}) = c_{j}(x_{j}- x_{j-1})