Gustave Choquet's 1954 paper "Theory of Capacities" introduces a systematic study of capacities, which are measures of the "size" of sets in a topological space. The paper begins with a problem posed by M. Brelot and H. Cartan: whether the interior and exterior Newtonian capacities of a Borel subset of $ \mathbb{R}^3 $ are equal. Choquet addresses this by studying non-additive set functions and their properties, particularly focusing on the Newtonian capacity, which satisfies a strong sub-additivity inequality. This inequality is shown to be equivalent to a condition involving the difference of capacities, and it leads to the conclusion that Borel and analytic sets are capable with respect to Newtonian capacity. The paper then extends this result to capacities associated with Green's functions and other classical capacities. The paper is divided into six chapters. Chapter I defines Borel and analytic sets in topological spaces, emphasizing the role of $ K_{\sigma\delta} $ sets. Chapter II discusses Newtonian and Greenian capacities, defining interior and exterior capacities and their properties. Chapter III introduces alternating and monotone functions and capacities, along with their relationships. Chapter IV explores the extension and restriction of capacities, showing how they can be used to regularize classes of sets. Chapter V studies operations on capacities and provides examples of capacities, including those derived from probabilistic schemes and energy functions. Chapter VI presents fundamental theorems on capacitability, showing that certain classes of sets are capable with respect to alternating capacities. Chapter VII examines extremal elements of convex cones and their integral representations, leading to a probabilistic interpretation of capacities. The paper establishes that capacities, such as Newtonian and Greenian capacities, satisfy strong sub-additivity properties and are closely related to functions with alternating signs in their successive derivatives. These properties allow for the construction of integral representations of capacities and provide a framework for studying their behavior on various classes of sets, including Borel and analytic sets. The work has had a significant impact on the theory of capacities and has been foundational in the development of modern potential theory and measure theory.Gustave Choquet's 1954 paper "Theory of Capacities" introduces a systematic study of capacities, which are measures of the "size" of sets in a topological space. The paper begins with a problem posed by M. Brelot and H. Cartan: whether the interior and exterior Newtonian capacities of a Borel subset of $ \mathbb{R}^3 $ are equal. Choquet addresses this by studying non-additive set functions and their properties, particularly focusing on the Newtonian capacity, which satisfies a strong sub-additivity inequality. This inequality is shown to be equivalent to a condition involving the difference of capacities, and it leads to the conclusion that Borel and analytic sets are capable with respect to Newtonian capacity. The paper then extends this result to capacities associated with Green's functions and other classical capacities. The paper is divided into six chapters. Chapter I defines Borel and analytic sets in topological spaces, emphasizing the role of $ K_{\sigma\delta} $ sets. Chapter II discusses Newtonian and Greenian capacities, defining interior and exterior capacities and their properties. Chapter III introduces alternating and monotone functions and capacities, along with their relationships. Chapter IV explores the extension and restriction of capacities, showing how they can be used to regularize classes of sets. Chapter V studies operations on capacities and provides examples of capacities, including those derived from probabilistic schemes and energy functions. Chapter VI presents fundamental theorems on capacitability, showing that certain classes of sets are capable with respect to alternating capacities. Chapter VII examines extremal elements of convex cones and their integral representations, leading to a probabilistic interpretation of capacities. The paper establishes that capacities, such as Newtonian and Greenian capacities, satisfy strong sub-additivity properties and are closely related to functions with alternating signs in their successive derivatives. These properties allow for the construction of integral representations of capacities and provide a framework for studying their behavior on various classes of sets, including Borel and analytic sets. The work has had a significant impact on the theory of capacities and has been foundational in the development of modern potential theory and measure theory.