support of a measurable function How we define support of a measurable function? Can we say that if $f$ is a measurable function with support $E$ then $E\cup F$ is also $supp(f)$, where $m(F)=0$ and $f=0$ on $F.$ 
 A: If I am guessing correctly, you would like to avoid unpleasant situations (e.g., non-uniqueness) with sets of measure zero.  So, how about this little construct for the case when $f$ is defined (and real-valued) on a measure space $(X, {\cal B}, \mu)$, where $X$ is a topological space and ${\cal B}$ the Borel algebra.  Assume, furthermore, $f \geq 0$ a.e..
Define the support of a continuous non-negative function $\phi : X \rightarrow [0, +\infty[$, as usual, as the closure of the set $\phi \neq 0$.
We will say that a continuous function $\phi : X \rightarrow [0, +\infty[$ majorizes $f$ if $\phi \geq f$ a.e..
We can now define the support of $f$ as the intersection of the supports of all the continuous $\phi : X \rightarrow [0, +\infty[$ that majorize $f$.
One resulting property is: since the supports of all these majorizing functions are closed, their intersection is also closed.
If $f$ is not non-negative a.e., then deal separately with its positive and negative parts, the way I described above.
