Fix an epsilon > 0. Let {r_k} be an enumeration of the rationales, and let X_k b

Question

Fix an epsilon > 0. Let {r_k} be an enumeration of the rationales, and let X_k be the characteristic function of the open interval (r_k - epsilon/2^k, r_k + epsilon/2^k). Define an increasing sequence of functions {s_n} by s_n = sigma^n _k = 1 X_k. Show that {s_n}is a Cauchy sequence of step functions which converges almost everywhere to an upper function f. Show that the integral of the upper function f equals 2 epsilon. Show that f is unbounded on every open subinterval of the reals. Finally, show that -f is not an upper function.

Explanation / Answer

a step function to be a function that is piecewise constant,

f(x)=i=1nci[ai,bi)

a function f:IRR

admissible, i.e. if it is the uniform limit of step functions.

Let =1/n,

nN. For we determine a >0 such that for all x,y[a,b] with

|xy|<,

we have

|f(x)f(y)|<.

Let m m the greatest integer with

<ba/ m or a+m<b.

For every integer l with 0lm we set

xl:=a+l and xm+1:=b

and define a step function by

fn(x)={f(xl), pentru xlx<xl+1f(b), pentru x=b.

Then for all x[a,b] we have

|f(x)fn(x)|=|f(x)f(xl)|<1/n,

as |xxl|< . Finally, (fn)nN converges uniformly to f.

Recall that a function is supposed to be (properly) Riemann integrable if for all >0 there exists a partition such that UL<. Suppose f becomes unbounded (say, unbounded above) near the point x0. How are we to make sense of the upper sum U when one of the intervals in the partition (the one containing x0) has no supremum. The definition only makes sense when f is bounded. Nonetheless, if f becomes unbounded near x0, it still may be (improperly) Riemann integrable on an interval containing x0. This is because we define the improper integral to be the limit of the proper integrals over the regions where we delete an open neighborhood of x0 , as we let that neighborhood get smaller and smaller.

If someone describes a function as "Riemann integrable" on a set S, it may be ambiguous if they mean properly or improperly (technically they should mean the former), and one has to infer from context whether they are implying that the function is bounded.

a function f:IRR

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