# measurable function with chat gpt

"please let me know the meaning of "measurable function" as easily as you can."

"A measurable function is a mathematical function that maps inputs to outputs, where the inputs and outputs are both sets, and the function preserves certain properties of those sets. In other words, a measurable function is a function that can be analyzed using tools from measure theory, which is a branch of mathematics that deals with the concepts of size and extent of sets.

For example, a function that maps real numbrers to real numbers is measurable if it preserves the measurability of the input set, which means that the preimage of any measurable set under the function is also measurable. Measurable functions are important in many areas of mathematics, including probability theory, functional analysis, and harmonic analysis, where they are used to study various properties of functions and their relationships with other mathematical objects.

こう帰ってきた。隙あらば応用先の話をしてくる。

つまり、可測関数の肝は、”入力”と”出力”に集合が入り、なおかつ入力の集合の”可測性”なるものを、出力の集合にも保存してくれる、ような関数のこと？

もう少しわかりやすく説明してもらうことにする。

”your explanation is difficult for me. please give me milder explanation and example of it"

Sure, let me try to explain it in simpler terms.

A measurable function is a function that preserves the "measurable" structure of sets. When we say a set is "measurable", it means we can assign it a size or extent using a mathematical concept called a "measure". For example, the length of a line segment or the area of a rectangle can be considered as measures.

Now, imagine we have a function that takes inputs from a measurable set and gives outputs in another measurable set. We say that this function is measurable if the preimage of any measurable set in the output set is a measurable set in the input set. This means that if we can measure the size or extent of a set in the output set, we can also measure the size or extent of the corresponding set in the input set.

For example, let's consider a function f(x) that maps real numbers to real numbers. We say that f(x) is measurable if the preimage of any measurable set A in the output set (the real numbers) is a measurable set in the input set (also the real numbers). In other words, if we can measure the size of a subset of real numbers, say [0,1], we can also measure the size of the corresponding subset of real numbers in the input set that maps to it under f(x).

Measurable functions have many applications in mathematics, such as in probability theory, where they are used to define probability distributions, and in analysis, where they are used to study the properties of functions and their relationships with other mathematical objects.

かなり腑に落ちた。

そもそも、測度というものは、集合に定義された追加の構造で、

２番目の黄色の通り、

つまり、何かしらの曲線Aから、別の曲線への写像Bがあって、

A, Bの上に長さが定義できて、Bの長さの取れる、どんな区分についても、それに対応するAの長さの取れる区分がある、という状況。（←良い説明か？）