In mathematics, a measure is a generalization of the concepts of length, area and volume. It is the central concept in measure theory and a key component of integration theory.

Measures can take on negative values, which leads to a number of interesting special cases, such as the Liouville and Gibbs measures.

## Axioms

Axioms are fundamental statements about real numbers or geometric figures. Some of them are also known as algebraic postulates. For example, the parallel axiom says that only one line can be drawn parallel to another through a point outside of it. Other axioms, such as the multiplication and division axioms, say that any figure can be multiplied or divided by any other figure, and that the results are always equal.

A mathematician uses a set of axioms to define a theory. These are not empirical, but they form a framework from which other theorems can be derived. They can be either logical or non-logical. Logical axioms are taken to be true within the system of logic they describe and are often shown in symbolic form. Non-logical axioms are genuine substantive declarations about the elements of a particular mathematical theory, such as arithmetic.

Mathematicians try to construct a set of axioms that are consistent, so that they do not contradict each other. However, in practice they do not always succeed.

## Units

A unit is an established reference allowing you to define the magnitude of a physical quantity. The length of a leg, for example, is measured in ‘pencil measures’, while the weight of a product is expressed in kilograms and tonnes (Metric) or ounces and pounds (Imperial).

Units are defined on a scientific basis and overseen by governmental or independent agencies. They are artifact-free, meaning they are not tied to a physical object that can be deteriorated or destroyed over time. They can be multiplied, compared and converted by applying a number of conversion factors.

The seven base units of the metric system are kilogram, metre, candela, volt, ampere, kelvin and mole. The metric system allows for easy multiplication between different quantities with the same base unit, such as metres and centimetres. For example, one kilometre equals 100 centimetres and vice versa. This makes it easier to compare measurements and make accurate calculations. This is a very important aspect of measurement.

## Uncertainty

Uncertainty is the standard deviation of a state-of-knowledge probability distribution over the possible values that could be attributed to a measured quantity. It is sometimes referred to as measurement error, but this term is more correctly used when describing the systematic errors caused by bias and other factors that affect all measurements and not just those made using a particular instrument.

Measurements have uncertainties, and it is important to understand how to evaluate and report them. Without uncertainty information, it is impossible to compare one measurement result with another and determine if they agree “apples to apples.”

All measurements incorporate some level of uncertainty regardless of the precision or accuracy of the measuring instrument. This uncertainty is due to the limitations of the instrument (systematic error), the skill of the experimenter making the measurement (random error) and other factors such as the environment and the sample being measured. The measurement result may or may not lie within the uncertainty that was determined, but it is highly unlikely to fall outside of this range.

## Measurement

The measurement of physical quantities is necessary for many important activities. Without it we could not construct buildings, use modern microwaves, or maintain accurate temperatures in refrigerators.

Early measurement theorists formulated axioms about the qualitative empirical structures that must be present for numerical representations to be meaningful. They used these axioms to prove theorems about the adequacy of addition (and other operations such as multiplication and division) in relation to magnitudes that exhibit these structures. These magnitudes Campbell called “fundamental”.

More recently, scholars have developed a range of realist theories of measurement. These can be grouped into two broad strands: information-theoretic accounts and model-based accounts. Some works do not fit neatly into either strand, and there is much ongoing debate.