A non-measurable set is a concept in mathematics, particularly in the field of measure theory. It refers to a set that cannot be assigned a meaningful size or measure using standard methods. For example, the Lebesgue measure, which is commonly used to determine the size of subsets of real numbers, cannot assign a value to certain sets, such as the Vitali set. Non-measurable sets challenge our intuitive understanding of size and volume. They arise in various mathematical contexts, illustrating the limitations of traditional measurement techniques. Understanding these sets helps mathematicians explore deeper properties of numbers and functions, contributing to the broader field of real analysis.