Search results
Results from the WOW.Com Content Network
When () <, a set of functions (,,) is uniformly integrable if and only if it is bounded in (,,) and has uniformly absolutely continuous integrals. If, in addition, μ {\displaystyle \mu } is atomless, then the uniform integrability is equivalent to the uniform absolute continuity of integrals.
The Riesz–Fischer theorem also applies in a more general setting. Let R be an inner product space consisting of functions (for example, measurable functions on the line, analytic functions in the unit disc; in old literature, sometimes called Euclidean Space), and let {} be an orthonormal system in R (e.g. Fourier basis, Hermite or Laguerre polynomials, etc. – see orthogonal polynomials ...
Then the function f(x) defined as the pointwise limit of f n (x) for x ∈ S \ N and by f(x) = 0 for x ∈ N, is measurable and is the pointwise limit of this modified function sequence. The values of these integrals are not influenced by these changes to the integrands on this μ-null set N , so the theorem continues to hold.
If X = [a,b] ⊆ R and μ is Lebesgue measure, there are sequences (g n) of step functions and (h n) of continuous functions converging globally in measure to f. If f and f n (n ∈ N) are in L p (μ) for some p > 0 and (f n) converges to f in the p-norm, then (f n) converges to f globally in measure. The converse is false.
Continuous functions, monotone functions, step functions, semicontinuous functions, Riemann-integrable functions, and functions of bounded variation are all Lebesgue measurable. [2] A function f : X → C {\displaystyle f:X\to \mathbb {C} } is measurable if and only if the real and imaginary parts are measurable.
Under the same hypothesis of the abstract Severini–Egorov theorem suppose that A is the union of a sequence of measurable sets of finite μ-measure, and (f n) is a given sequence of M-valued measurable functions on some measure space (X,Σ,μ), such that (f n) converges μ-almost everywhere on A to a limit function f, then A can be expressed ...
Download as PDF; Printable version; In other projects ... two measurable functions in L p in terms of the L p-norms of those functions ... : X → R be measurable ...
Virtual instrument software architecture (VISA) is a widely used application programming interface (API) in the test and measurement (T&M) industry for communicating with instruments from a computer. VISA is an industry standard implemented by several T&M companies, such as, Anritsu , Bustec , Keysight Technologies , Kikusui, National ...