2 Operations on measurable functions Theorem 2.1 Let f : E → IR and g : E → IR be measurable functions and let k ∈ IR. Example 1. First we show that A Sis a ˙-algebra on X. Measurable functions are characterized as pointwise limits of finite valued functions.

For each , set 1) Each function is pointwise the limit of a sequence of simple functions.

Example 2. As \(C=E^{2},\) a complex function \(f : S \rightarrow C\) is simple, elementary, or measurable on \(A\) iff its real and imaginary parts are. It is weaker than uniform convergence, to which it … However, if \(\mathcal{M}\) is a \(\sigma\)-ring, one can make the limit uniform. Any non-negative measurable function is the pointwise limit of a monotonic increasing sequence of non-negative simple functions. n} be a sequence of measurable functions on E that converges pointwise a.e. A more serious positive indicator of the reasonable-ness of Borel-measurable functions as a larger class containing continuous functions: [1.3] Theorem: Every pointwise limit of Borel-measurable functions is Borel-measurable.

Exercise 4. Since each Ais a ˙-algebra, Ac2Afor every A2F. (a) ffngn2N is Cauchy in measure.

Indeed, we have the following theorem. Let fn be measurable functions on X. Theorem 5. Let {f n} be the sequence of functions on R defined by f n(x) = nx. 2) The pointwise limit of a sequence of real valued measurable functions is measurable. A continuous function pulls back open sets to open sets, while a measurable function pulls back measurable sets to measurable sets. (b) Show that f(x) = 8 <: lim n!1 fn(x); if the limit exists; 0; otherwise; is a measurable function.
Basically it states that every measureable function can be written as the pointwise limit of a sequence of simple functions. Measurable functions and simple functions The class of all real measurable functions on (Ω,A) ... n=1 in E whose pointwise limit is f. Proof. This sequence does not converge pointwise on R because lim n→∞ f n(x) = ∞ for any x > 0. for every x in D, is called the pointwise limit of the sequence {f n}.

Indeed, let be a non-negative measurable function defined over the measure space as before.
The choice of -algebras in the definition above is sometimes implicit and left up to the context. In addition, the function f/g is measurable if g(x) 6= 0 for all x ∈ E. Proof: We will only prove that f+g is measurable as the others will be left as exercises. Some authors define measurable functions as exclusively real-valued ones with respect to the Borel algebra.

Given measurable fn on X, the following statements are equivalent. By Definition \(4,\) a measurable function is a pointwise limit of elementary maps. Almost everywhere convergence. measurable functions for which we will seek to define the Lebesgue integral. In measure theory, one talks about almost everywhere convergence of a sequence of measurable functions defined on a measurable space.That means pointwise convergence almost everywhere, i.e. For each , subdivide the range of into intervals, of which have length . Jensen’s inequality for the integration of convex functions and Egorov’s theorem saying that an almost everywhere converging sequence of functions also converges almost uniformly, i.e. Proof. Let E 0 ⊂ E for which m(E 0) = 0 and {f n} converges to f pointwise on E \E 0.