## Example 7.4.10(d): Properties of the Lebesgue Integral

Suppose

Define the sets
*f*is a bounded, non-negative function defined on a measurable set*with finite measure such that***E***. Show that*_{E}f(x) dx = 0*f*must then be equal to zero except on a set of measure zero.ThenE_{n}= { x E: f(x) 1/n }

= { x E: f(x) # 0 }Z

*and*

**E**_{n}=**Z***. Using the previous two examples we get:*

**E**_{n}**E**so that0 =_{E}f(x) dx_{En}f(x) dx 1/n m(E_{n})

*m(*for all

**E**_{n}) = 0*n*. But then

which is what we had to prove.m() = m(ZE_{n})E_{n}= 0