In mathematics, the capacity of a set in Euclidean space is a measure of the "size" of that set. Unlike, say, Lebesgue measure, which measures a set's volume or physical extent, capacity is a mathematical analogue of a set's ability to hold electrical charge. More precisely, it is the capacitance of the set: the total charge a set can hold while maintaining a given potential energy. The potential energy is computed with respect to an idealized ground at infinity for the harmonic or Newtonian capacity, and with respect to a surface for the condenser capacity.

The notion of capacity of a set and of "capacitable" set was introduced by Gustave Choquet in 1950: for a detailed account, see reference (Choquet 1986).

Let Σ be a closed, smooth, (n − 1)-dimensional hypersurface in n-dimensional Euclidean space ${\displaystyle \mathbb {R} ^{n}}$, n ≥ 3; K will denote the n-dimensional compact (i.e., closed and bounded) set of which Σ is the boundary. Let S be another (n − 1)-dimensional hypersurface that encloses Σ: in reference to its origins in electromagnetism, the pair (Σ, S) is known as a condenser. The condenser capacity of Σ relative to S, denoted C(Σ, S) or cap(Σ, S), is given by the surface integral

where:

C(Σ, S) can be equivalently defined by the volume integral

The condenser capacity also has a variational characterization: C(Σ, S) is the infimum of the Dirichlet's energy functional

over all continuously differentiable functions v on D with v(x) = 1 on Σ and v(x) = 0 on S.

Heuristically, the harmonic capacity of K, the region bounded by Σ, can be found by taking the condenser capacity of Σ with respect to infinity. More precisely, let u be the harmonic function in the complement of K satisfying u = 1 on Σ and u(x) → 0 as x → ∞. Thus u is the Newtonian potential of the simple layer Σ. Then the harmonic capacity or Newtonian capacity of K, denoted C(K) or cap(K), is then defined by