is just a special case of the more general -space (defined below). Lp spaces and Lebesgue integrals An space may be defined as a space of measurable functions for which the -th power of the absolute value is Lebesgue integrable, where functions which agree almost everywhere are identified. More generally, let and be a measure space. Consider the set of all measurable functions from to or whose absolute value raised to the -th power has a finite integral, or equivalently, that For the space is the space of measurable functions bounded almost everywhere, whose seminorm is the infimum of (the absolute values of) these bounds, which when is the same as the essential supremum of its absolute value: Two functions and defined on are said to be , written , if the set is measurable and has measure zero. If is a measurable function that is equal to almost everywhere then for every and thus for all Seminormed space of -th power integrable functions Each set of functions forms a vector space when addition and scalar multiplication are defined pointwise. That the sum of two -th power integrable functions and is again -th power integrable follows from although it is also a consequence of Minkowski's inequality which establishes that satisfies the triangle inequality. That is closed under scalar multiplication is due to being absolutely homogeneous, which means that for every scalar and every function Absolute homogeneity, the triangle inequality, and non-negativity are the defining properties of a seminorm. Thus is a seminorm and the set of -th power integrable functions together with the function defines a seminormed vector space. In general, the seminorm is not a norm because there might exist measurable functions that satisfy but are not equal to ( is a norm if and only if no such exists). Quotient vector space Like every seminorm, the seminorm induces a norm (defined shortly) on the quotient of by the vector subspace This normed quotient space is called and it is the subject of this article. If the seminorm happens to be a norm then the normed quotient space that will now be defined is linearly isometrically isomorphic to they will be, up to a linear isometry, the same normed space and so they may both be called " space". If is any measurable function, then if and only if almost everywhere. Since the right hand side ( a.e.) does