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Similarly, a function g defined on domain D and having the same codomain (K, ≤) is an upper bound of f, if g(x) ≥ f (x) for each x in D. The function g is further said to be an upper bound of a set of functions, if it is an upper bound of each function in that set.
There is a corresponding greatest-lower-bound property; an ordered set possesses the greatest-lower-bound property if and only if it also possesses the least-upper-bound property; the least-upper-bound of the set of lower bounds of a set is the greatest-lower-bound, and the greatest-lower-bound of the set of upper bounds of a set is the least ...
Then the supremum of is = if the set of upper bounds is nonempty, and = + otherwise. Now assume in addition that ( X , Σ , μ ) {\displaystyle (X,\Sigma ,\mu )} is a measure space and, for simplicity, assume that the function f {\displaystyle f} is measurable .
In mathematics, the least-upper-bound property (sometimes called completeness, supremum property or l.u.b. property) [1] is a fundamental property of the real numbers. More generally, a partially ordered set X has the least-upper-bound property if every non-empty subset of X with an upper bound has a least upper bound (supremum) in X .
A continuous function () on the closed interval [,] showing the absolute max (red) and the absolute min (blue).. In calculus, the extreme value theorem states that if a real-valued function is continuous on the closed and bounded interval [,], then must attain a maximum and a minimum, each at least once.
A real set with upper bounds and its supremum. A set S of real numbers is called bounded from above if there exists some real number k (not necessarily in S) such that k ≥ s for all s in S. The number k is called an upper bound of S. The terms bounded from below and lower bound are similarly defined. A set S is bounded if it
Markov's inequality (and other similar inequalities) relate probabilities to expectations, and provide (frequently loose but still useful) bounds for the cumulative distribution function of a random variable. Markov's inequality can also be used to upper bound the expectation of a non-negative random variable in terms of its distribution function.
Every function f belonging to L p (Ω), 1 ≤ p ≤ +∞, where Ω is an open subset of , is locally integrable. Proof. The case p = 1 is trivial, therefore in the sequel of the proof it is assumed that 1 < p ≤ +∞. Consider the characteristic function χ K of a compact subset K of Ω: then, for p ≤ +∞,