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Measure Theory with respect to Soap Bubbles - 1

Minimal surfaces, otherwise better known in the form of a Mobius strip, sphere, or a helicoid, are inclined to provide the least surface area with respect to some constraint(s). We want to see a connection to Measure Theory, if one exists at all (and it does). To drive our motivation further, we pick the principle example of a soap bubble. 

But first, what is Measure Theory? It is built on the concepts researched by Henri Lebesgue. Lebesgue measure theory consists of two key concepts used here- Lebesgue Measure and Lebesgue Integral and stands on the fundamentals of measure of sets and intervals developed by Lebesgue. 

We can think of the concept of measure coming from the length between two coordinates or a length of an interval on coordinate plane such that l(I)= b-a

So where is the minimal surface here? It would not precisely exist in the coordinate system as definite as above is, but I promise, we will be tempted to draw some common strings once we get acquainted with measure and functions and integrals. 

A measure may be called as a Lebesgue measure generally if the following properties are met:

  1. Non-negativity: m(A) ≥ 0
  2. Null set has measure zero i.e., m(ϕ) = 0
  3. Countable additivity: If E is a sequence of disjoint sets, then the union of sequence of disjoint sets equals the sum of measure of each set. This means that the measure of the countable union of these sets is equal to the sum of the measures of the individual sets.
A set E is said to be Lebesgue measurable or briefly measurable if for each set A we have   
m*(A)=m*(AՈE) + m*(AՈE').

This equality becomes useful in finding whether a set A is measurable or not. Here m* denotes the outer measure. Here m*(AՈE') is required to account for any error in our approximation such that we can include it in our measure. 

This equation is the necessary condition for a set or an interval to be measurable, i.e., if A is measurable then it will satisfy this equation, but merely satisfying this criterion does not guarantee the set’s measurability. 

Moving on to the second key concept- functions. Functions that are continuous and are defined on measurable sets are measurable as continuous functions map open intervals to open intervals and the preimage of any open set under a continuous function is also an open set. 

A real valued function f defined on a measurable set E is said to Lebesgue measurable or measurable if the set {x∈ E: f(x) > a} is measurable, i.e., we can say that a function defined on a measurable set is measurable. A notable instance to understand this is of Dirichlet’s function. 

For g:R→R, the Dirichlet function g: [0, 1] ~ R defined by the piecewise function g(x),

g(x)= 1 , x is rational

g(x)= 0 , x is irrational

The function g is bounded and measurable on [0, 1] and hence it is Lebesgue integrable. We can show its measurability by showing that g(x) > a for different values of a in the interval [0,1]. Let's take different cases for a.

For a < 0: We get g(x) > a

For 0 < a < 1: For a rational x, we have g(x)=1 which is greater than a.

For a ≥ 1: Since g(x) can never be greater than 1,{x: g(x) > a} = ∅. The empty set is measurable. 

Overall, the Dirichlet function g(x) is measurable on the interval [0,1] because the sets defined by the condition g(x) > a are all measurable for any real number a. However, it is worth noting that while this function is Lebesgue integrable, it is not Reimann integrable due its discontinuity (we'll circle back to this later).

Lastly, we will touch upon the Lebesgue Integral. This integral can be considered as a generalization of the Reimann integral and can be applied to a broader range of functions in infinite dimensional spaces. Lebesgue integral overcomes the drawbacks posed by the R-integral and provides more powerful convergence theorems under less restrictive conditions.

A bounded function f is said to be Lebesgue integral over a set E if 

Here the LHS is the lower Lebesgue integral and RHS is the upper Lebesgue Integral. The common value is called the Lebesgue Integral and is denoted as L ∫ f(x) dx. (To be able to attain this integral, the lower and upper integrals must converge to the same value.)






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