Lebesgue Measure Homework: Clarifying Royden's Definition

[EV33]

Homework Statement

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 [itex]\alpha[/itex] the set
1.) {x: f(x)>[itex]\alpha[/itex]} is measurable
2.) {x: f(x)<[itex]\alpha[/itex]} is measurable
3.) {x: f(x)≤[itex]\alpha[/itex]} is measurable
4.) {x: f(x)≥[itex]\alpha[/itex]} 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.

 

[lippyka]

Homework Equations N/AThe Attempt at a Solution 1st Question: Yes, if a function is Lebesgue Measurable then its domain is measurable and all four of these statements are true. 2nd Question: No, when we say a function is Lebesgue Measurable this does not necessarily mean that its image is measurable. The image of a function may not be measurable even if the function itself is Lebesgue Measurable.