Set Functions

A real-valued function on X, f\colon X\to\mathbf{R}, assigns elements of the set X to real numbers in \mathbf{R}. A real-valued set function σ\colon \mathcal{P}(X)\to\mathbf{R} assigns subsets of X to real numbers in \mathbf{R}, where the power set \mathcal{P}(X) is the set of all subsets of X. If σ(E\cup F) = σ(E) + σ(F) - σ(E\cap F), E,F\subseteq X, and σ(\emptyset) = 0 then σ is a measure. Measures don’t count things twice and the measure of nothing is zero.

Newton’s fundamental theorem of calculus demonstates how finding tangent lines to the graph of a function on real numbers is related to the area underneath its graph: \int_a^b f'(x)\,dx = f(b) - f(a). Special cases of finding areas bounded by curves had been solved prior to Newton but he was the first to invent a systematic treatment, aka calculus, for doing this. After a long train of brilliant mathematicians following the scent Newton layed down we now have the Generalized Stokes Theorem \int_Ω dω = \int_{∂Ω} ω where Ω is a region of n-dimensional space, ω is a smooth form on Ω, ∂Ω is the boundary of Ω, and dω is the differential of ω.

Newton’s theorem is the 1-dimensional case where Ω = [a,b] and ω = f(x) is differentiable function. In this case dω = f'(x)\,dx is a 1-form and ∂[a,b] = δ_b - δ_a is the difference of delta functions at the endpoints of the interval. We will not delve furthur into the defintions and conditions required for this theorem, but hopefully whet your appetite to learn more with another example.

Suppose dx\,dy = -dy\,dx. This implies dx\,dx = 0 when y = x and makes it plausible that d^2x = 0. Let P(x,y) and Q(x,y) be functions on \mathbf{R}^2 and write P_x = ∂P/∂x for partial derivatives. If ω = P\,dx + Q\,dy is a 1-form then dω = d(P\,dx + Q\,dy) = dP\,dx + dQ\,dy = (P_x\,dx + P_y\,dy)dx + (Q_x\,dx + Q_y\,dy)dy = P_y\,dy\,dx + Q_y\,dx\,dy = (Q_y - P_x)dx\,dy is a 2-form. Green’s theorem is the 2-dimensional case of Stokes theorem \int_Ω (Q_y - P_x)dx\,dy = \int_{∂Ω} P\,dx + Q\,dy.

Further reading: Grassmann, R. C. Buck.

A different train of brilliant thought led to considering how to extend the set function on intervals [a,b]\mapsto \int_a^b f to all sets and for functions f that might not be differentiable. This led to many complications if the set function is required to be countably additive. Integrating any bounded function over a finitely additive measure is elementary.

Write Y^X for the set of all functions from X to Y. We can identify the power set \mathcal{P}(X) with 2^X = \{0,1\}^X where S\subseteq X corresponds to the characteristic function 1_S with 1_S(x) = 1 if x\in S and 1_S(x) = 0 if x\not\in S.

Real-valued functions on X, \mathbf{R}^X, are a vector space with pointwise addition (f + g)(x) = f(x) + g(x) and scalar multiplication (af)(x) = a f(x), f,g\in\mathbf{R}^X, a\in\mathbf{R}. The dual of a vector space V is the set of all functions v^*\in\mathbf{R}^V\colon V\to\mathbf{R} that are linear, v^*(av + w) = av^*(v) + v^*(w), a\in\mathbf{R} and v, w\in V. The dual pairing is a function \langle , \rangle\colon V\times V^*\to\mathbf{R} defined by \langle v, v^*\rangle = v^*(v), v\in V, v^*\in V^*.

A function f is bounded if there is a number M\in\mathbf{R} with |f(x)| \le M for all x\in X. The smallest number satifying this is \|f\|, the norm of f. The set of bounded functions on X is a normed vector space, B(X). A norm on a vector space is a function \|\cdot\|\colon V\to\mathbf{R} that satisfies \|v\|\ge 0, \|av\| = |a| \|v\| and \|v + w\| \le \|v\| + \|w\|, v,w\in V, a\in\mathbf{R}. Norms induce a metric d(v, w) = \|v - w\| that provides a topology on V. A linear transformation T\colon V\to W is continuous in the metric topology if and only if it is bounded, \|Tv\| \le \|T\|\|v\|.

The dual of a normed vector space V is the set of all function \mathbf{R}^V that are linear and continuous/bounded.