Given a continuous function $X(t)$ mapping $[0,1]$ continuously onto $[0,1]$, several properties are given which this function must fulfill if there is to be another continuous function $Y(t)$ so that $(X(t),Y(t))$ takes the unit interval onto the unit square.

Upper and lower limits for the length of the graph of one to one, onto, Baire one functions from the unit interval to itself are shown to be infinity and one, respectively. Such functions having graphs of large dimension and non-measurable functions are also considered.

Convergence in measure of a sequence of functions was defined by F. Riesz in 1909.(Cf.[5] and [2] p.177 ff.)It was used to provide a proof of the completeness of L p (Riesz-Fischer Theorem).It is also used in the proof of the weak law of large numbers and thus is sometimes called convergence in probability.A sequence of real valued functions {f n } defined on a measure space is said to converge in measure to a function f , writtenand all functions are presumed measurable and defined on a measure space X.Standard results involving limits in measure can be found in [4].In particular, convergence in measure does not imply convergence at a single point.However, the limit is unique (up to sets of measure 0); the limit of the sum of two sequences which converge in measure is the sum of the limits in measure; the limit in measure of a constant times the functions in a sequence is the constant times the limit function; and the limit in measure of a sequence of measurable functions is measurable.In Munroe [4] it is proved that if mX < ∞ then the limit in measure of the product of two sequences is the product of the limits providing the limits in measure exist.To show that this does not hold on spaces of infinite measure, he gives the example: f n (x) = x and g n (x) = 1/n for each x ∈ (0, ∞) where m is Lebesgue measure on the line.However, it is not difficult to show that the limit in measure of the product is the product of the limits under certain conditions.In particular, if the functions f and g are either bounded or Lebesgue integrable and ifActually, there is a more general condition which guarantees that the limit in measure of the product is the product of the limits in measure.It is given

An \(s\)-set in Euclidean space is a set of finite, non-zero, Hausdorff \(s\)-dimensional measure. Call an \(s\)-set straight if its \(s\)-measure agrees with its Method I \(s\)-outer measure. Examples are given where there is a continuous, one-to-one function \(f\) on \(\mathbb{R}^n\) which is measure preserving on \( E\) so that \(f(E)\) is straight (such an \(f\) will be called a straightening of \(E\)). It is shown that any \(s\)-set can be written as a countable union of sets for which there are straightenings.

a) Sketch the graph of y = f ( x 2 ) , indicating the coordinates of the stationary point and the coordinates of the points of intersection of the graph with the x-axis. (3) b) Figure 2 shows a sketch of the graph having one of the following equations with an appropriate value of either p, q or r. y = f(x + p), where p is a constant y = f(x) + q , where q is a constant y = rf(x), where r is a constant