6.2 Relative Measurability
6.2.1 Relative Measure of Sets
Given a set 
, 
being the σ-field of the Borel subsets and μ the Lebesgue measure on the real line 
, the relative measure of A is defined as (Kac and Steinhaus 1938), (Le
kow and Napolitano 2006)
provided that the limit exists. If the limit in (6.2) exists, it is independent of t0 and the set A is said to be relatively measurable. From definition (6.2) it follows that the relative measure of a set is the Lebesgue measure of the set normalized to that of the whole real line. Thus, sets with finite Lebesgue measure have zero relative measure, and only sets with infinite Lebesgue measure can have nonzero relative measure. In (Lekow and Napolitano 2006), the following properties of the relative measure are proved: The class of RM sets is not closed under union and intersection; The relative measure μR is additive, but not σ-additive; If A is a RM set, then 
is RM. Moreover, the lack of σ-additivity does not allow to prove the continuity of μR. In (Lekow and Napolitano 2006) it is also shown that non RM sets are “not so rare or sophisticated” as non-Lebesgue-measurable sets. In fact, non RM sets can be easily constructed and visualized.
The lack of σ-additivity proved for the relative measure μR, in contrast with the σ-additivity assumed for the probability P in the classical stochastic approach, constitutes one of the motivations of using the functional approach in signal analysis. In fact, due to such a difference between μR and P, results holding for stochastic processes do not necessarily have a counterpart in terms of functions of time representing sample paths of these stochastic processes.
6.2.2 Relatively Measurable Functions
Let 
be a Lebesgue measurable real-valued function (or signal). The function x(t) is said to be relatively measurable if and only if the set 
is RM for every 
, where Ξ0 is an at most countable set of points. Each RM function x(t) generates a function
in all points ξ where the limit exists. In (6.3),
is the indicator of the set 
(note the difference with definition (1.2)). The function Fx(ξ) defined in (6.3) has values in [0, 1] and is non decreasing. Thus, it has all the properties of a distribution function, except the right-continuity in the discontinuity points. Furthermore, as for every bounded nondecreasing function, the set of discontinuity points is at most countable. The function Fx(ξ) represents the fraction-of-time probability that the signal x(t) is below the threshold ξ (Gardner 1987d) and hence is named fraction-of-time distribution function.
Since non relatively measurable sets can be easily constructed (unlike non Lebesgue measurable sets), the lack of relative measurability of a function is not a rare property as the lack of Lebesgue measurability.
The function Fx(ξ) allows to define all familiar probabilistic parameters such as mean, variance, moments, and cumulants. Furthermore, if x(t) is a RM, not necessarily bounded, function and g(ξ) satisfies appropriate regularity conditions, then the following fundamental theorem of expectation can be proved (Lekow and Napolitano 2006, Theorem 3.2):
where the second integral is in the Riemann-Stiltjes sense. That is, the infinite-time average is the expectation operator of the distribution (6.3)
The set of RM functions can be shown to be not closed under addition and multiplication operations. Consequently, RM functions do not constitute a vector space. On the contrary, the linear combination of two stochastic processes is still a stochastic process, provided that the two sample spaces are assumed to be jointly measurable, which is an implicitly made assumption.
6.2.3 Jointly Relatively Measurable Functions
In (Lekow and Napolitano 2006), the concept of joint relative measurability between functions is introduced. It is shown that operations on jointly RM functions can lead to a RM function. Therefore, for such a function, a probabilistic model based on time averages can be constructed. The joint relative measurability is an analytical property of functions and, hence, easier to be verified than the analogous property in the stochastic process framework, that is, the joint measurability of sample spaces. The latter property, in fact, cannot be easily verified in applications since, generally, the sample spaces are not specified.
Two Lebesgue measurable functions x(t) and y(t) are said to be jointly RM (Lekow and Napolitano 2006) if the limit
exists for all 
, where Ξ0 is an at most countable set of lines of 
. The function Fyx has all the properties of a bivariate joint distribution function except the right continuity in the discontinuity points.
Let x(t) and y(t) be jointly RM functions. Then both x(t) and y(t) are RM. Moreover, the sum x(t) + y(t) and the product x(t)y(t) are RM, provided that at least one of the functions is bounded. In contrast, if y(t) is not RM, then y(t) is not jointly RM with any RM x(t) (Lekow and Napolitano 2006), (Lekow and Napolitano 2007).
If x(t) is RM, then the lag product x(t)x(t + τ) is not necessarily RM 
. However, if x(t) and its shifted version x(t + τ) are jointly RM, then the lag product x(t)x(t + τ) is RM.
The notion of joint relative measurability can be extended to the multidimensional case. A finite collection of Lebesgue measurable functions x1(t), ..., xn(t) is jointly RM if the limit
exists for all 
, where Ξ0 is at most a countable set of (n − 1)-dimensional hyperplanes of 
.
The function 
has all the properties of a nth-order joint distribution function, except the right continuity property with respect to each of the ξ1, ..., ξn variables in the discontinuity points.
Let x1(t + τ1), ..., xn−1(t + τn−1), xn(t) be not necessarily bounded functions jointly RM for any τ1, τ2, ..., τn−1 and let g(ξ1, ..., ξn) a bounded function satisfying appropriate regularity conditions. Then, for any 
the following fundamental theorem of expectation for the multivariate case holds (Lekow and Napolitano 2006, Theorem 4.5)
where the first integral is in the Lebesgue sense and the second is in the Riemann-Stieltjes sense and
Consequently, if x1(t + τ1), ..., xn−1(t + τn−1), xn(t) are bounded jointly RM functions for any τ1, τ2, ..., τn−1, then their temporal cross moment can be expressed as
In the special case of n = 2, if x(t) and y(t) are bounded functions and and y(t + τ) and x(t) are jointly RM for any τ, then the cross-correlation function of x and y is given by
where
That is, the cross-correlation function (6.12a) is the expected value corresponding to the distribution (6.13).
As a consequence of the lack of σ-additivity of the relative measure μR, the corresponding expectation operator, the infinite-time average, is linear, but not σ-linear. The infinite-time average of the linear combination of a finite number of jointly RM functions (with at least one of them bounded) is equal to the linear combination of the time averages. This is not always true if we have a countable infinity of functions of time. For example, the periodic function cos 2(t) has infinite-time average equal to 1/2. However, 
and the time average of cos 2(t)1[k,k+1)(t) is zero. This result is different form the corresponding one in the stochastic approach where the expectation operator is σ-linear, provided that the underlying infinite series of random variables is absolutely convergent (Kolmogorov 1933).
Finally, note that the case of complex-valued signals can be treated with obvious changes by considering the joint characterization of real and imaginary parts (which leads to a doubling of the order of the distributions).
6.2.4 Conditional Relative Measurability and Independence
The definition of independence between two signals in the functional approach is given starting from the definition of conditional relative measurability. Then the result that the joint distribution function of two independent signals factorizes into the product of the two marginal distributions is obtained as a theorem and an intuitive concept of independence is shown to correspond to such a mathematical property.
Let A and B be Lebesgue measurable sets and {Bn} be an arbitrary increasing sequence of Lebesgue measurable subsets of B with 0< μ(Bn) < ∞, such that m < n ⇒ Bm ⊆ Bn; 
; and 0< lim nμ(Bn)/n < ∞. The conditional relative measure μR(· |B) of the set A given B is defined as (Lekow and Napolitano 2006)
provided that the limit exists. Note that in definition (6.14) the two sets A and B can be such that neither of the two is RM, nor A ∩ B is RM, but μR(A|B) exists (Lekow and Napolitano 2006, Example 5.1).
Let the sets A and B be such that μR(A|B) exists and A is RM. The sets A and B are said to be independent if
Let x(t) be a RM function and 
. Let y(t) be a Lebesgue measurable function and 
. Assume also that 
, where Ξ0 is at most a countable set of lines, μR(A|B) exists. The signals x(t) and y(t) are called independent if (6.15) holds 
.
Assume that x(t) and y(t) are jointly RM. The signals x(t) and y(t) are independent, if and only if, 
except at most a countable set of lines, it results that (Lekow and Napolitano 2006, Theorem 5.2)
(6.16) 
where Fxy(ξ1, ξ2) is the joint distribution function of x(t) and y(t) in the sense of definition (6.7) and Fx(ξ1) and Fy(ξ2) are the distribution functions of x(t) and y(t), respectively, in the sense of definition (6.3).
The definition of independence of two signals x(t) and y(t) based on (6.15) leads to the following intuitive interpretation of the concept of independence. If x(t) and y(t) are independent, then defined the sets 
and 
, we have μR(A|B) = μR(A). That is, the normalization in (6.14) of the measure of the set A, constructed from x(t), made by subsets Bn of the set B, constructed from y(t), gives rise to the same result obtained considering the normalization Bn = [− n/2, n/2]. In other words, the function y(t) from which the normalizing sets Bn are constructed, has no influence on the relative measure μR(A|B). Therefore, according with the intuitive concept of independence, the two functions or signals x(t) and y(t) have no link each other. Note that such an intuitive interpretation of independence has no counterpart in the stochastic process approach where independence of processes is defined as the factorization of the joint distribution function into the product of the marginal ones (Kolmogorov 1933), (Doob 1953), (Billingsley 1968).
6.2.5 Examples
Signals of interest that can be characterized in the functional approach are almost-periodic functions and pseudorandom functions. The latter are appropriate models for realizations of several digital communications signals. They are shown to be RM and characterized in (Lekow and Napolitano 2006).
In (Lekow and Napolitano 2006), it is shown that if x(t) is a uniformly almost-periodic function (Section 1.2.1) then the limit (6.3) does not necessarily exist at the discontinuity points of Fx(ξ). The function of t, 1{x(t)≤ξ} is discontinuous in t for any ξ such that the limit (6.3) exists. Moreover, it is shown that for any ξ such that the limit (6.3) exists, the function of t, 1{x(t)≤ξ} is a Wp-AP function (Definition 1.2.4). Analogously, it can be shown that the function of t, 1{x(t)≤ξ} is a Bp-AP function. In (Lekow and Napolitano 2006) uniformly AP functions are proved to be RM and jointly RM. Moreover they are proved to be equal in distribution to the asymptotically AP functions which are obtained by adding 
terms to the uniformly AP functions.
In (Kac 1959, p. 52), it is shown that the functions x(t) = cos (λ1t) and y(t) = cos (λ2t), with λ1 and λ2 incommensurate, are independent.
Examples of non-RM functions are provided in (Lekow and Napolitano 2006) and (Lekow and Napolitano 2007). These functions exhibit statistical functions defined in terms of time averages that are not convergent. Thus they can be suitably exploited to design secure communications systems where an unauthorized user cannot discover the modulation format by second-and higher-order (cyclic) spectral analysis of the transmitted signal.