measurable set

简明释义

可测集

英英释义

例句

1.The concept of a measurable set allows us to assign a size or measure to subsets of a given space.

可测集 的概念使我们能够为给定空间的子集分配大小或度量。

2.To prove a function is integrable, we often need to show that its domain is a measurable set.

为了证明一个函数是可积的，我们通常需要展示它的定义域是一个 可测集。

3.A measurable set can be used to define integrals in the context of Lebesgue integration.

可测集 可以用于在勒贝格积分的背景下定义积分。

4.In probability theory, a measurable set is essential for defining events and calculating probabilities.

在概率论中，可测集 对于定义事件和计算概率至关重要。

5.In real analysis, every open interval is a measurable set in the standard Lebesgue measure.

在实分析中，每个开区间都是标准勒贝格度量下的 可测集。

作文

In the realm of mathematics, particularly in measure theory, the concept of a measurable set plays a pivotal role. A measurable set is defined as a subset of a given space that can be assigned a meaningful size or measure. This concept is crucial for understanding various mathematical constructs, including integration and probability. To illustrate this, let us consider the real number line. The intervals on this line, such as [0, 1] or (2, 5), are examples of measurable sets. Each of these intervals has a well-defined length, which is a measure that we can calculate easily. The significance of measurable sets extends beyond mere definitions; they form the foundation upon which we build our understanding of functions and their integrals. For instance, when we talk about integrating a function over an interval, we are essentially summing up the values of that function over a measurable set. Without the concept of measurable sets, it would be challenging to define what we mean by the 'area under the curve.' Moreover, the notion of measurable sets is not limited to simple geometric shapes. In more complex spaces, like those encountered in functional analysis, measurable sets can take on intricate forms. For example, consider the space of all continuous functions on a closed interval. The collection of all continuous functions that fall within a certain range can also be considered a measurable set in a higher-dimensional context. One of the most fascinating aspects of measurable sets is their relationship with probability theory. In probability, we often deal with events that can be represented as measurable sets. For instance, if we roll a die, the event of rolling an even number can be represented as the measurable set {2, 4, 6}. The probability of this event occurring can then be calculated using the measure of this set relative to the total outcomes. However, not every set is measurable. There are certain sets, known as non-measurable sets, that cannot be assigned a measure in a consistent way. The existence of such sets often leads to paradoxes and challenges in mathematical reasoning. This highlights the importance of defining measurable sets carefully and understanding their properties. In conclusion, the concept of a measurable set is fundamental in various branches of mathematics, particularly in measure theory and probability. It allows mathematicians and statisticians to assign sizes to sets, facilitating the study of functions, integrals, and probabilities. Understanding measurable sets not only enhances our mathematical knowledge but also equips us with the tools necessary to tackle complex problems in both theoretical and applied contexts.

在数学领域，特别是在测度理论中，可测集的概念起着至关重要的作用。可测集被定义为可以赋予有意义的大小或测度的给定空间的子集。这个概念对于理解各种数学构造，包括积分和概率，至关重要。为了说明这一点，让我们考虑实数线。该线上的区间，例如[0, 1]或(2, 5)，是可测集的示例。这些区间每个都有一个明确的长度，这是我们可以轻松计算的测度。可测集的重要性超越了简单的定义；它们构成了我们建立对函数及其积分理解的基础。例如，当我们谈论在一个区间上对一个函数进行积分时，我们实际上是在对该函数在一个可测集上的值进行求和。如果没有可测集的概念，定义“曲线下的面积”将是一个挑战。此外，可测集的概念并不限于简单的几何形状。在更复杂的空间中，例如在泛函分析中遇到的空间，可测集可能会呈现出复杂的形式。例如，考虑在闭区间上所有连续函数的空间。所有落在某个范围内的连续函数的集合也可以视为高维上下文中的可测集。可测集最迷人的一方面是它们与概率理论的关系。在概率中，我们经常处理可以表示为可测集的事件。例如，如果我们掷骰子，掷出偶数的事件可以表示为可测集 {2, 4, 6}。然后，可以使用该集合相对于总结果的测度来计算此事件发生的概率。然而，并不是每个集合都是可测的。有些集合被称为非可测集，无法以一致的方式赋予测度。这类集合的存在通常会导致悖论和数学推理中的挑战。这突显了仔细定义可测集及理解其性质的重要性。总之，可测集的概念在数学的各个分支中，特别是在测度理论和概率中是基础性的。它使数学家和统计学家能够为集合赋予大小，从而促进对函数、积分和概率的研究。理解可测集不仅增强了我们的数学知识，而且为我们提供了解决理论和应用背景中复杂问题所需的工具。