# Homework Help: Lebesgue measure

1. Apr 16, 2012

### EV33

1. The problem statement, all variables and given/known data
I just have a few quick questions about the definition of Lebesgue Measure of a function ( I just want some clarification on what I read in Royden)

In Chapter 3 of Royden, An extended real valued function is defined as being Lebesgue measurable if its domain is measurable and if it satisfies one of the following:
For each real number $\alpha$ the set
1.) {x: f(x)>$\alpha$} is measurable
2.) {x: f(x)<$\alpha$} is measurable
3.) {x: f(x)≤$\alpha$} is measurable
4.) {x: f(x)≥$\alpha$} is measurable

1st Question: In a proposition before this definition in my book it says that if the domain of an extended real valued function is measurable then statements 1 through 4 are equivalent (the statements above). So does this mean that if a function is Lebesgue Measurable that its domain is measurable and that all 4 of the above statements hold? I am thinking yes since one of them must hold but they are all 4 equivalent.

2nd Question: When we say a function is Lebesgue Measurable does this also mean that its image is measurable?

Thank you for your time.