Let be Lebesgue integrable functions. Let . Show that is well defined.
i am not sure what it means by a function is well defined. would someone tell me what i have to show to say that it is well defined? any help would be appreciated.
Well-defined means that for each input of a function, there is one output. For example, a circle is not a well-defined function.
First, note that since are Lebesgue integrable, they are well defined.
Also, if and only if .
Then, we have that since are well-defined, then , on account of . Similarly, .
Hence , and is well-defined.