Measure and Probability Lebesgue Integration (Cont’) 1. Let µ be...

Question
Measure and Probability
Lebesgue Integration (Cont’)
1. Let µ be a measure defined on the Borel σ - algebra of R. Assume that µ(R) < ∞. Let f(x) be a bounded continuous on R function. Prove that the function
g(t)=
∫f (tx)µ(dx),
R
is continuous on R).
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      • Illuminate\Cache\Events\CacheHit (272μs)
      • Illuminate\Cache\Events\CacheHit (266μs)
      • Illuminate\Cache\Events\CacheHit (191μs)
      • Illuminate\Cache\Events\CacheHit (169μs)
      • Illuminate\Cache\Events\CacheHit (194μs)
      • Illuminate\Cache\Events\CacheHit (326μs)
      • Illuminate\Cache\Events\CacheHit (1.57ms)
      • Illuminate\Cache\Events\CacheHit (206μs)
      • Illuminate\Cache\Events\CacheHit (208μs)
      • Illuminate\Cache\Events\CacheHit (380μs)
      • Illuminate\Cache\Events\CacheHit (361μs)
      • Illuminate\Cache\Events\CacheHit (202μs)
      • Illuminate\Cache\Events\CacheHit (316μs)
      • Illuminate\Cache\Events\CacheHit (187μs)
      • creating: site.layouts.app (575μs)
      • composing: site.layouts.app (26μs)
      • creating: components.canonical (513μs)
      • composing: components.canonical (97μs)
      • creating: components.open-graph (300μs)
      • composing: components.open-graph (107μs)
      • creating: site.headers.header (328μs)
      • composing: site.headers.header (87μs)
      • Illuminate\Cache\Events\CacheHit (2.31ms)
      • creating: components.footer (76μs)
      • composing: components.footer (88μs)
      • Illuminate\Cache\Events\CacheHit (1.08ms)
      • Illuminate\Cache\Events\CacheMissed (294μs)
      • Illuminate\Database\Events\StatementPrepared (712μs)
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      • eloquent.retrieved: App\Models\SubjectCat (153μs)
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      • Illuminate\Cache\Events\CacheHit (283μs)
      • Illuminate\Cache\Events\CacheHit (197μs)
      • Illuminate\Cache\Events\CacheHit (185μs)
      • Illuminate\Cache\Events\CacheHit (496μs)
      • Illuminate\Cache\Events\CacheHit (336μs)
      • Illuminate\Cache\Events\CacheHit (733μs)
      • Illuminate\Cache\Events\CacheHit (430μs)
      • Illuminate\Cache\Events\CacheHit (521μs)
      • Illuminate\Cache\Events\CacheHit (210μs)
      • Illuminate\Cache\Events\CacheHit (195μs)
      • Illuminate\Cache\Events\CacheHit (212μs)
      • Illuminate\Cache\Events\CacheHit (206μs)
      • creating: components.forms.contact-us (439μs)
      • composing: components.forms.contact-us (161μs)
      • creating: components.forms.get-started (247μs)
      • composing: components.forms.get-started (70μs)
      • creating: components.forms.free-tool-download (98μs)
      • composing: components.forms.free-tool-download (68μs)
      • creating: components.forms.claim-free-worksheet (102μs)
      • composing: components.forms.claim-free-worksheet (175μs)
      • creating: components.forms.tutor-subscription-waitlist (133μs)
      • composing: components.forms.tutor-subscription-waitlist (70μs)
      • creating: components.forms.tutor-subscription-join (103μs)
      • composing: components.forms.tutor-subscription-join (72μs)
      • creating: components.forms.tutor-support (102μs)
      • composing: components.forms.tutor-support (63μs)
      • 312 x Illuminate\Cache\Events\CacheHit (19.6%)
        128ms
        19 x Illuminate\Database\Events\StatementPrepared (5.44%)
        35.49ms
        19 x Illuminate\Database\Events\QueryExecuted (4.47%)
        29.17ms
        10 x Illuminate\Cache\Events\CacheMissed (1.57%)
        10.25ms
        10 x Illuminate\Cache\Events\KeyWritten (1.51%)
        9.88ms
        1 x Illuminate\Foundation\Events\LocaleUpdated (0.77%)
        5.04ms
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        11 x eloquent.retrieved: App\Models\SubjectCat (0.17%)
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        1 x Illuminate\Database\Events\ConnectionEstablished (0.15%)
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        1 x Illuminate\Routing\Events\RouteMatched (0.09%)
        602μs
        1 x creating: site.layouts.app (0.09%)
        575μs
        1 x creating: components.canonical (0.08%)
        513μs
        1 x creating: components.forms.contact-us (0.07%)
        439μs
        1 x eloquent.booting: App\Models\SubjectCat (0.05%)
        350μs
        1 x creating: site.headers.header (0.05%)
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        7 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (0.05%)
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        1 x creating: components.open-graph (0.05%)
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        275μs
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        247μs
        13 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.03%)
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        1 x composing: components.breadcrumbs (0.03%)
        180μs
        1 x composing: components.forms.claim-free-worksheet (0.03%)
        175μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibrary (0.03%)
        172μs
        1 x composing: components.forms.contact-us (0.02%)
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        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.02%)
        144μs
        1 x composing: homework.show (0.02%)
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        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibrary (0.02%)
        135μs
        1 x creating: components.forms.tutor-subscription-waitlist (0.02%)
        133μs
        1 x eloquent.booting: App\Models\Subject (0.02%)
        109μs
        1 x composing: components.open-graph (0.02%)
        107μs
        1 x creating: components.forms.tutor-subscription-join (0.02%)
        103μs
        1 x creating: components.forms.tutor-support (0.02%)
        102μs
        1 x creating: components.forms.claim-free-worksheet (0.02%)
        102μs
        1 x creating: components.forms.free-tool-download (0.02%)
        98μs
        1 x composing: components.canonical (0.01%)
        97μs
        1 x composing: components.footer (0.01%)
        88μs
        1 x composing: site.headers.header (0.01%)
        87μs
        1 x creating: components.footer (0.01%)
        76μs
        1 x composing: components.forms.tutor-subscription-join (0.01%)
        72μs
        1 x composing: components.forms.tutor-subscription-waitlist (0.01%)
        70μs
        1 x composing: components.forms.get-started (0.01%)
        70μs
        1 x composing: components.forms.free-tool-download (0.01%)
        68μs
        1 x composing: components.forms.tutor-support (0.01%)
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        1 x eloquent.booted: App\Models\Subject (0.01%)
        43μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.01%)
        42μs
        1 x eloquent.booted: App\Models\SubjectCat (0.01%)
        40μs
        1 x composing: site.layouts.app (0%)
        26μs
      14 templates were rendered
      • 1x homework.showshow.blade.phpblade
      • 1x components.breadcrumbsbreadcrumbs.blade.phpblade
      • 1x site.layouts.appapp.blade.phpblade
      • 1x components.canonicalcanonical.blade.phpblade
      • 1x components.open-graphopen-graph.blade.phpblade
      • 1x site.headers.headerheader.blade.phpblade
      • 1x components.footerfooter.blade.phpblade
      • 1x components.forms.contact-uscontact-us.blade.phpblade
      • 1x components.forms.get-startedget-started.blade.phpblade
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      • 1x components.forms.claim-free-worksheetclaim-free-worksheet.blade.phpblade
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      uri
      GET college-homework-library/{category}/{subject}/{id}
      middleware
      web, utm.parameters
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      App\Http\Controllers\HomeworkLibraryController@show
      namespace
      where
      as
      homework.show
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      19 statements were executed, 5 of which were duplicates, 14 unique. Show only duplicated44.16ms
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        Backtrace
        • 13. app/Http/Controllers/HomeworkLibraryController.php:91
        • 14. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 15. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 16. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
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        Backtrace
        • 16. app/Http/Controllers/HomeworkLibraryController.php:97
        • 17. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 18. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 19. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 20. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `subjects` where `subjects`.`id` in (125)
        950μstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 24. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 25. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` in (22007)
        1.13mstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 24. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 25. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `subject_cats` where `subject_cats`.`id` = 1 limit 1
        1.04mstwenty4_siteHomeworkLibrary.php#201
        Bindings
        • 0: 1
        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
        • 26. app/Http/Controllers/HomeworkLibraryController.php:105
        • 27. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 28. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 29. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
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        2.52mstwenty4_siteHomeworkLibraryRepository.php#30
        Bindings
        • 0: 22007
        • 1: 125
        • 2: published
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        Backtrace
        • 14. app/Repositories/HomeworkLibraryRepository.php:30
        • 15. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 16. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 18. app/Repositories/HomeworkLibraryRepository.php:39
        • 19. app/Http/Controllers/HomeworkLibraryController.php:139
      • select * from `subjects` where `subjects`.`id` in (125)
        1.07mstwenty4_siteHomeworkLibraryRepository.php#30
        Backtrace
        • 19. app/Repositories/HomeworkLibraryRepository.php:30
        • 20. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 21. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 23. app/Repositories/HomeworkLibraryRepository.php:39
        • 24. app/Http/Controllers/HomeworkLibraryController.php:139
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 22007 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'question' order by `order` asc, `id` asc
        930μstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 22007
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        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:58
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
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        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:69
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
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        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:80
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
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        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
        • 32. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
        • 33. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
        • 34. vendor/livewire/livewire/src/ComponentConcerns/RendersLivewireComponents.php:69
        • 35. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:70
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        Bindings
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        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
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        3.07mstwenty4_siteSubject.php#100
        Bindings
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        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
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        1.4mstwenty4_siteSubject.php#100
        Bindings
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        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 10 limit 1
        1.36mstwenty4_siteSubject.php#100
        Bindings
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        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 33 limit 1
        1.26mstwenty4_siteSubject.php#100
        Bindings
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        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 11 limit 1
        10.04mstwenty4_siteSubject.php#100
        Bindings
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        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
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        Bindings
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        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 34 limit 1
        2.92mstwenty4_siteSubject.php#100
        Bindings
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        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 36 limit 1
        1.05mstwenty4_siteSubject.php#100
        Bindings
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        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 28. view::components.footer:170
      App\Models\HomeworkLibrary\HomeworkLibraryFile
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      App\Models\SubjectCat
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      App\Models\HomeworkLibrary\HomeworkLibrary
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