Uncertainty

How do we measure continuous quantities?

First, set up some sort of "triangulation" of the space in which the continuous quantity can take values. For example, a ruler is a "triangulated" line. Typically some sort of quantitative system is chosen to keep track of the cells; e.g. for a ruler we count how many ticks there are.

A measurement of a continuous quantity is simply an identification of which one of those intervals the continuous quantity lies inside. It's a declaration "this quantity lives inside this interval."

How do we apply functions to continuous quantities?

Motivating example

Let's suppose that we are measuring lengths with a meter stick, which goes down to the nearest 2 millimeters. That is, the ticks on the stick appear at 0mm, 2mm, 4mm, 6mm, etc.

Suppose that we measure a certain length, then, by some process, we double the length. That is, we apply the "continuum" function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = 2x} .

That is what Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} does to length itself, but what does Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} do to measurements of length?

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} denote a length lying somewhere in the range Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [2k-2, 2k]} millimeters (a minimal interval on the meter stick). Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = 2x} lies somewhere in the range Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [4k-4, 4k] = [4k - 4, 4k -2] \cup [4k - 2, 4k]} , which is the union of two minimal intervals of the meter stick. So maybe we should say that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} induces a multi-valued function?

Hmm, but what would be the induced function of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g(x) = 1.5x} ? It would send the interval Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [2k-2, 2k]} to the interval Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [3k-3, 3k]} , which is not necessarily a union of two minimal intervals.

I think it would be best to treat the induced function in a probabilistic manner: If the length Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is equally likely to be anywhere in the range Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [2k-2, 2k]} , then

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2x} it has a 1/2 chance of being in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [4k - 4, 4k -2]} , and a 1/2 chance of being in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [4k - 2, 4k]} .
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3x} has,
  • if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} is even, a 2/3 chance of being in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [3k - 2, 3k]} , and a 1/3 chance of being in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [3k - 4, 3k - 2]} .
  • if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} is odd, a 2/3 chance of being in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [3(k+1) - 6, 3(k+1)-4]} , and a 1/3 chance of being in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [3(k+1) - 4, 3(k+1)-2]} .

The general principle is that you have some sort of probability measure on the triangulation, and then you push it forward to another one.

The "problem" with this is that the true probability distributions are "continuum," as the following example demonstrates.

Let the grid on one axis consist of intervals Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [n \pi / 6, (n+1)\pi / 6]} , let the grid on the other axis consist of intervals Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [0.1n, 0.1(n+1)]} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\in \mathbb{Z}} , and let the continuum function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h(x) = \cos(x)} . Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h} sends the interval Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [0, \pi/6]} to the interval Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\sqrt{3}/2,1] = [0.866\cdots, 1] \subset [0.8, 0.9] \cup [0.9, 1.0]} So after applying Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h} to a length in that internal, we will get a length which is in the interval Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [0.9, 1.0]} with probability Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \frac{1}{10 - 5 \sqrt{3}} \approx 0.75 } , and in the interval Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [0.8, 0.9]} with probability Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \frac{9 - 5 \sqrt{3} }{10 - 5 \sqrt{3}} \approx 0.25 } .

General formulation

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f : \mathbb{R} \rightarrow \mathbb{R}} be a continuous function (which is also a continuum function), and suppose we have a grid consisting of intervals Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a_n, b_n]} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n, b_n} are real numbers such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b_n = a_{n + 1}} . We refer to the ordered set of these intervals as "the grid" and denote it as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma} .

Definition. A probability distribution on a set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} is a function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p: S \rightarrow [0,1]} , such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_s = 0 } for all but finitely many Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s \in S} , and $\sum _{s\in S}p_{s}=1$. Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Prob}(S)} denote the set of probability distributions on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} .

Since $f$ is continuous, it sends intervals to intervals. If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I = [a,b]} is any interval, define its volume as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \text{Vol}(I):= b - a} . Then $f$ induces a function ${\widetilde {f}}:\Gamma \rightarrow {\text{Prob}}(\Gamma )$, where

$${\widetilde {f}}(n)_{m}:={\frac {{\text{Vol}}([a_{n},b_{n}]\cap f^{-1}([a_{m},b_{m}]))}{{\text{Vol}}([a_{n},b_{n}])}}$$

The interpretation of ${\widetilde {f}}$ is completely clear: If the function $f$ is applied to a quantity that was measured as being in the grid interval $[a_{n},b_{n}]$, then the probability that the resulting quantity will be measured as being in the grid interval $[a_{m},b_{m}]$ is ${\widetilde {f}}(n)_{m}$.

We can say that "a measurement of $f$ relative to the triangulation $\Gamma$" is a field of random variables, one at each triangle, where the random variable at triangle $n$ is drawn from the probability distribution ${\widetilde {f}}(n)$.

Measure-theoretic interpretation

For my purposes in this section, by a measure on $\mathbb {R}$ I mean one for which all points have measure 0. $\Gamma$ is a countable cover of $\mathbb {R}$ by measurable sets. Any measure $\mu$ on $\mathbb {R}$ restricts to a measure $\mu |_{\Gamma }$ on $\Gamma$, where ${\textstyle \mu |_{\Gamma }(S):=\mu \left(\bigcup _{s\in S}s\right)=\sum _{s\in S}\mu (s).}$ In particular, if $\mu$ is a probability measure then so is $\mu |_{\Gamma }$.

For an interval $I_{n}$ of $\Gamma$, let $\mu _{n}$ be the probability measure defined by ${\textstyle \mu _{n}(A):={\frac {{\text{Vol}}(A\cap I_{n})}{{\text{Vol}}(I_{n})}}}$. Let $f:\mathbb {R} \rightarrow \mathbb {R}$ be any measurable function. Then $f$ induces a probability measure $f_{*}\mu _{n}$ on $\mathbb {R}$, and hence a probability measure $f_{*}\mu _{n}|_{\Gamma }$ on $\Gamma$. In this language, ${\widetilde {f}}$ is simply the function sending

$${\begin{aligned}{\widetilde {f}}:\Gamma &\rightarrow {\text{Prob}}(\Gamma )\\I_{n}&\mapsto (f_{*}\mu _{n})|_{\Gamma }\end{aligned}}.$$

One can easily check that the formulas agree.

Renormalization

Let $f:\mathbb {R} \rightarrow \mathbb {R}$ be some continuum function. There is a relationship between $f$ as measured on a fine triangulation, and $f$ as measured on a coarse triangulation.

Motivating example

Let's suppose that we are measuring lengths with two meter sticks, one which goes down to the nearest 2 millimeters, and one which goes down to the nearest 1 millimeter. That is, the ticks on the first stick appear at 0mm, 2mm, 4mm, 6mm, etc., and the ticks on the second stick appear at 0mm, 1mm, 2mm, 3mm, etc. The former stick defines a triangulation which we will call $\Gamma$, and the latter defines a triangulation which we will call $\Gamma '$.

There is a (discontinuous!!) function $i:\Gamma '\rightarrow \Gamma$, sending $[0,1]\mapsto [0,2]$, $[1,2]\mapsto [0,2]$, $[2,3]\mapsto [2,4]$, $[3,4]\mapsto [2,4]$, etc. This function is the answer to the question: "If a quantity is measured in a given grid cell of the fine triangulation, which grid cell of the course triangulation will it be measured in?"

More generally, the grid cells of the fine triangulation might not be contained inside the grid cells of the coarse triangulation, so a quantity with a given fine-scale measurement might have multiple possible coarse-scale measurements. So really the answer to the italicized question above should not be a function $\Gamma '\rightarrow \Gamma$, but rather should be a function $\Gamma '\rightarrow {\text{Prob}}(\Gamma )$.