Let be a sequence of -measurable real-valued functions on a measure space . Then the functions
are also -measurable.
I already have proven this statement and my question is about something else. In the script it says that the reader may use the hint:
For a sequence of real numbers the following is true:
Now, I have not used this in my proof, but I can assure you, that my proof is airtight nonetheless. Anyway, can someone point out why this holds? I have not seen this equality before and did not know that one can present like that. I know that for a set of limit points we can say that and vice versa. I do not even see how this property can be useful for our proof.