{"id":184926,"date":"2025-01-21T18:52:02","date_gmt":"2025-01-21T18:52:02","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=184926"},"modified":"2025-01-21T18:52:04","modified_gmt":"2025-01-21T18:52:04","slug":"circular-lebesgue-measure","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/01\/21\/circular-lebesgue-measure\/","title":{"rendered":"Circular Lebesgue measure"},"content":{"rendered":"\n

Circular Lebesgue measure. Define to be the unit circle in the complex plane. Define<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
\"\"<\/figure>\n\n\n\n

The correct answer and explanation is:<\/mark><\/strong><\/p>\n\n\n\n

The unit circle<\/strong> in the complex plane, C\\mathbb{C}, is defined as: S1={z\u2208C:\u2223z\u2223=1},S^1 = \\{z \\in \\mathbb{C} : |z| = 1\\},<\/p>\n\n\n\n

where \u2223z\u2223=Re(z)2+Im(z)2|z| = \\sqrt{\\text{Re}(z)^2 + \\text{Im}(z)^2} is the modulus of zz.<\/p>\n\n\n\n

The circular Lebesgue measure<\/strong> on S1S^1 is a measure \u03bc\\mu that assigns “lengths” to subsets of S1S^1, analogous to how the Lebesgue measure assigns “lengths” to subsets of R\\mathbb{R}. Here\u2019s the correct definition and explanation:<\/p>\n\n\n\n

Measure Definition:<\/h3>\n\n\n\n

For a Borel set E\u2282S1E \\subset S^1, the circular Lebesgue measure \u03bc(E)\\mu(E) is defined by: \u03bc(E)=\u222bE1\u2009d\u03b8,\\mu(E) = \\int_E 1 \\, d\\theta,<\/p>\n\n\n\n

where \u03b8\\theta is the angle parameterizing points on S1S^1 in polar coordinates, z=ei\u03b8z = e^{i\\theta} for \u03b8\u2208[0,2\u03c0)\\theta \\in [0, 2\\pi).<\/p>\n\n\n\n

Explanation:<\/h3>\n\n\n\n

The circular Lebesgue measure on S1S^1 arises as the natural counterpart to the standard Lebesgue measure on [0,2\u03c0)[0, 2\\pi), but restricted to the unit circle. Since S1S^1 is compact and has no endpoints (it’s a closed loop), this measure is defined in terms of the angular coordinate \u03b8\\theta, where each point z\u2208S1z \\in S^1 can be written as z=ei\u03b8z = e^{i\\theta}.<\/p>\n\n\n\n

Intuitively, the circular Lebesgue measure “counts” the length of an arc EE on S1S^1 based on its angular width in radians. For instance, the full circle S1S^1 corresponds to \u03b8\u2208[0,2\u03c0)\\theta \\in [0, 2\\pi), and the total measure is \u03bc(S1)=2\u03c0\\mu(S^1) = 2\\pi. This ensures the measure aligns with our geometric understanding of the circle’s circumference.<\/p>\n\n\n\n

For any measurable subset E\u2282S1E \\subset S^1, \u03bc(E)\\mu(E) equals the arc length of EE. If EE is a single arc, \u03bc(E)\\mu(E) is proportional to the angular span of EE. The measure is invariant under rotations (a property known as translation-invariance in R\\mathbb{R}), reflecting the circle’s symmetry.<\/p>\n\n\n\n

The circular Lebesgue measure is fundamental in Fourier analysis on S1S^1, where functions are studied in terms of their Fourier series expansions. It also appears in probability theory (e.g., uniform distribution on S1S^1) and physics, particularly when modeling periodic phenomena.<\/p>\n","protected":false},"excerpt":{"rendered":"

Circular Lebesgue measure. Define to be the unit circle in the complex plane. Define The correct answer and explanation is: The unit circle in the complex plane, C\\mathbb{C}, is defined as: S1={z\u2208C:\u2223z\u2223=1},S^1 = \\{z \\in \\mathbb{C} : |z| = 1\\}, where \u2223z\u2223=Re(z)2+Im(z)2|z| = \\sqrt{\\text{Re}(z)^2 + \\text{Im}(z)^2} is the modulus of zz. The circular Lebesgue measure […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-184926","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/184926","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=184926"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/184926\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=184926"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=184926"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=184926"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}