If u has a weak derivative, it is often written D ^ { \ alpha } u since weak derivatives are unique ( at least, up to a set of measure zero, see below ).
12.
It is sometimes said that the circle map maps the rationals, a set of measure zero at " K " = 0, to a set of non-zero measure for K \ neq 0.
13.
Then there is a ?-finite quasi-invariant measure ? on " X " which is unique up to measure equivalence ( that is any two such measures have the same sets of measure zero ).
14.
The integral makes sense because the set of directions where projection doesn't give a knot diagram is a set of measure zero and " n " ( " v " ) is locally constant when defined.
15.
Such functions are defined only up to a set of measure zero, and since the boundary \ partial \ Omega does have measure zero, any function in a Sobolev space can be completely redefined on the boundary without changing the function as an element in that space.
16.
The theorem is analogous to regular Fubini's theorem for the case where the considered function is a characteristic function of a set in a product space, with usual correspondences meagre set with set of measure zero, comeagre set with one of full measure, a set with Baire property with a measurable set.
17.
More specifically, a property holds almost everywhere if the set of elements for which the property does not hold is a set of measure zero ( Halmos 1974 ), or equivalently if the set of elements for which the property holds is complete, it is sufficient that the set is contained within a set of measure zero.
18.
More specifically, a property holds almost everywhere if the set of elements for which the property does not hold is a set of measure zero ( Halmos 1974 ), or equivalently if the set of elements for which the property holds is complete, it is sufficient that the set is contained within a set of measure zero.
19.
For every smooth test function \ phi \ in C _ c ^ \ infty ( \ Omega ) with compact support, then ( up to redefinition on a set of measure zero ) u \ in C ^ { \ infty } ( \ Omega ) is smooth and satisfies \ Delta u = 0 pointwise in \ Omega.
