PREMEASURE Meaning and
Definition
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A premeasure is a term used in mathematics, particularly in the field of measure theory, to describe a function that assigns a non-negative value to a collection of sets. More specifically, a premeasure is a set function defined on a certain class of sets, which may or may not satisfy the properties of a measure.
Formally, a premeasure is a function μ: Σ → [0, ∞) where Σ is a collection of sets, called a σ-algebra, such that μ(∅) = 0 (where ∅ denotes the empty set) and μ(A) ≤ μ(B) whenever A ⊆ B.
The key difference between a premeasure and a measure is that a premeasure does not guarantee countable additivity. In other words, the value assigned by a premeasure to the union of countably many disjoint sets may not be equal to the sum of individual values. Countable additivity is a desirable property in measure theory as it allows for simpler and more powerful mathematical analysis.
Premeasures play a crucial role in defining measures by utilizing an extension theorem, which extends the premeasure to a measure on a larger class of sets. Once extended, the measure inherits important properties such as countable additivity, enabling more rigorous and comprehensive analysis.
In summary, a premeasure is a set function that assigns non-negative values to a collection of sets, but does not necessarily satisfy countable additivity. It serves as a precursor to measures and provides a starting point for defining measures through extension theorems.
Etymology of PREMEASURE
The term "premeasure" is derived from the combination of the prefix "pre-" and the word "measure".
The prefix "pre-" comes from Latin and means "before" or "prior to". It is commonly used in English to indicate something that comes before or precedes another.
The word "measure" has its roots in Old French and Latin. In Old French, it was spelled "mesure", which came from the Latin word "mensura". Both the Old French and Latin forms have the same meaning, referring to the act or result of determining the size, extent, or quantity of something.
By combining the prefix "pre-" with the word "measure", the term "premeasure" is formed, suggesting a measure or action taken in advance or before another measure or action.
Conjugate verb Premeasure
CONDITIONAL
I
would premeasure
you
would premeasure
he/she/it
would premeasure
we
would premeasure
they
would premeasure
CONDITIONAL CONTINUOUS
I
would be premeasuring
you
would be premeasuring
he/she/it
would be premeasuring
we
would be premeasuring
they
would be premeasuring
CONDITIONAL PERFECT
I
would have premeasure
you
would have premeasure
he/she/it
would have premeasure
we
would have premeasure
they
would have premeasure
CONDITIONAL PERFECT CONTINUOUS
I
would have been premeasuring
you
would have been premeasuring
he/she/it
would have been premeasuring
we
would have been premeasuring
they
would have been premeasuring
FUTURE
I
will premeasure
you
will premeasure
he/she/it
will premeasure
we
will premeasure
they
will premeasure
FUTURE CONTINUOUS
I
will be premeasuring
you
will be premeasuring
he/she/it
will be premeasuring
we
will be premeasuring
they
will be premeasuring
FUTURE PERFECT
I
will have premeasured
you
will have premeasured
he/she/it
will have premeasured
we
will have premeasured
they
will have premeasured
FUTURE PERFECT CONTINUOUS
I
will have been premeasuring
you
will have been premeasuring
he/she/it
will have been premeasuring
we
will have been premeasuring
they
will have been premeasuring
IMPERATIVE
you
premeasure
we
let´s premeasure
to premeasure
PAST
I
premeasured
you
premeasured
he/she/it
premeasured
we
premeasured
they
premeasured
PAST CONTINUOUS
I
was premeasuring
you
were premeasuring
he/she/it
was premeasuring
we
were premeasuring
they
were premeasuring
PAST PARTICIPLE
premeasured
PAST PERFECT
I
had premeasured
you
had premeasured
he/she/it
had premeasured
we
had premeasured
they
had premeasured
PAST PERFECT CONTINUOUS
I
had been premeasuring
you
had been premeasuring
he/she/it
had been premeasuring
we
had been premeasuring
they
had been premeasuring
PRESENT
I
premeasure
you
premeasure
he/she/it
premeasures
we
premeasure
they
premeasure
PRESENT CONTINUOUS
I
am premeasuring
you
are premeasuring
he/she/it
is premeasuring
we
are premeasuring
they
are premeasuring
PRESENT PARTICIPLE
premeasuring
PRESENT PERFECT
I
have premeasured
you
have premeasured
he/she/it
has premeasured
we
have premeasured
they
have premeasured
PRESENT PERFECT CONTINUOUS
I
have been premeasuring
you
have been premeasuring
he/she/it
has been premeasuring
we
have been premeasuring
they
have been premeasuring
