Suppose that $F$ is a field of subsets of a set $U$, $N$ is a positive integer, ${\left\{{\alpha }_k\right\}}_{k=1}^N$ is a sequence of functions from $F$ into $\exp (\mathbf{R})$ and $\mu$ is a real, nonnegative - valued finitely additive function on $F$. Suppose that ${\mathrm{\Xi }}^N$ is the set of all $N$ - dimensional subintervals of ${\mathbf{R}}^N$. It is shown that there is a nonnegative - valued function A from ${\mathrm{\Xi }}^N\times F$ into $\mathbf{R}$ such that for each $V$ in $F$, $A(\cdot ,V)$ is finitely additive on ${\mathrm{\Xi }}^N$, such that if for $k=1,\dots ,N,{\alpha }_k$ is $\mu$ - summable (see “Fields of Sets, Set Functions, Set Function Integrals, and Finite Additivity”, Internat. J. Math. & Math. Sci., Vol. 7 No. 2 (1984) pp 209 - 233) and $g$ is a real - valued function on ${\mathbf{R}}^N$ satisfying certain continuity and boundedness conditions, then

$${\displaystyle \int \dots }{\displaystyle {\int }_R[g{\displaystyle {\int }_{U}A(\cdot ,\cdot )]\to {\sigma }_{\mu }(g({\alpha }_1,\dots ,{\alpha }_N))(U),}}$$

$$R=[H_1;K_1]\times \cdots \times [H_N;K_N],\min \{-H_1,\dots ,-H_N,K_1,\dots ,K_N\}\to \infty ,$$

where ${\sigma }_{\mu }$ is the $\mu$-summability operator and all integrals are refinement - wise limits of the appropriate sums.

We study the classical Chebyshev inequality

$${\int }_0^{a}x(s)ds{\int }_0^{a}y(s)ds\le a{\int }_0^{a}x(s)y(s)ds$$

and prove new variants, generalizations and abstractions. Among these are Chebyshev’s inequality for strongly increasing functions, positive convex and concave functions, and generalizations of the Ky Fan inequality. Our abstractions involve Chebyshev’s inequality in Banach function spaces and symmetric spaces. These considerations lead to generalizations of inequalities of Favard and Barnes. The constants that arise are studied and shown to be sharp in general.

Let $(P_j)j=1,2,3\dots$ be a sequence of sets $P_j$ of real numbers such that each $P_j$ is countable and has more than one point. Let $A$ denote the set of all numbers $x$ that can be uniquely expressed $x={\displaystyle {\sum }_{j=1}^{\infty }{a}_j(a_j\in P_j)}$. Let $B$ denote the set of numbers $y$ that can be expressed in at least two ways: $y={\displaystyle {\sum }_{j=1}^{\infty }{b}_j}={\displaystyle {\sum }_{j=1}^{\infty }{c}_j}\left(b_j\in P_j,c_j\in P_j\right)$ such that $b_j\ne c_j$ for at least one index $j$. Here we prove that if $A\cup B$ is a second category subset of $R$, then $A\cup B$ is a subset of the closure of $B$. In particular, if $A\cup B$ is a dense second category subset of $R$, then $B$ is dense in $R$; if $B$ is a nowhere dense subset of $R$, then $A$ is a first category subset of $R$. This unifies and generalizes results of M. Petkovsek [P] and of M. Starbird and T. Starbird [SS].

After extending Borel’s classification of measure zero sets in previous papers, we now look at some collections of measure zero sets naturally defined from properties of their covers and prove they actually form $\sigma$-ideals; we also study the parameters involved in their definitions, using some methods of Hausdorff measure introduced by Zakrzewski.