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Yes there are, and you must beware of assuming that a **function** is **integrable** without looking at it. The simplest examples of non-**integrable** **functions** are: in the interval [0, b]; and in any interval containing 0. These are intrinsically not **integrable**, because the area that their integral would represent is infinite. The wave **function** should be square **integrable**. The wave **function** must be single valued. It means for any given values of x and t, there should be a unique value of Ψ (x, t) so there is only a single value for the probability of the system being in a given state. It must have a finite value or it must be normalized. **Properties** of Wave **Function**. Homework 10: Show that a Riemann **integrable** **function** is Lebesgue **integrable** (the integral for the Lebesgue measure exists), and the values of the two integrals are the same. Hint: Turn sequences of upper and lower sums into sequences of integrals of step **functions**, and show that the sequences of step **functions** are Cauchy. Solution. Then, a predictable process is -**integrable** if and only if it is locally -**integrable**. Proof: If is locally -**integrable** then there exist stopping times such that . Choose bounded predictable processes which tend to zero as goes to infinity. Then, for any fixed , are dominated by . So, equation ( 4) and dominated convergence in probability give. of a locally **integrable** **function**. In this work, we focus on pointwise **properties** of **functions** outside exceptional sets of codimension one. The set of non-Lebesgue points of a classical Sobolev **function** is a set of measure zero with respect the Hausdor measure of codimension one and this holds true also in metric spaces supporting a doubling. **function**. It is clear that scalar multiplication takes a square-**integrable** **function** to another square-**integrable** **function**, so this de nition makes sense. You can check the usual addition of **functions** and multiplication of **functions** by scalars have all the required **properties** listed above. The zero vector is just the **function** that assigns the. The following proposition lists many elementary **properties** of measurable **functions** and the Lebesgue integral that we will prove in time. For the third property, two. The Lebesgue Integral 5 ... If fand gare Lebesgue **integrable** **functions** on Xand f g, then Z X fd Z X gd : 3. If f;gare measurable **function** on Xand f= galmost everywhere, then f.