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σ-algebras and measures
Which sets can be measured, and the countable additivity that makes limits work.
A σ-algebra is a family of sets closed under complements and countable unions; a measure assigns sizes additively over countable disjoint unions. Lebesgue measure gives intervals their length and extends to far stranger sets. Picture it: covering a set by intervals and taking the cheapest total length. Think it: not every set is measurable (Vitali), which is why the σ-algebra is part of the definition.
हल किया गया उदाहरण · integrate 1/10 dx from 2 to 5
चरण दर चरण
- \int_{2}^{5} \frac{1}{10}\, dx
First find an antiderivative F, then evaluate F(b) − F(a).
- \int \frac{1}{10}\, dx = \frac{x}{10}
The integral of a constant c is c·x.
- F(5) - F(2) = \left(\frac{1}{2}\right) - \left(\frac{1}{5}\right)
Fundamental theorem of calculus: plug in the limits.
- = \frac{3}{10}
Simplify.
जवाब दिखाएँ
अब आप कोई प्रश्न चुनें, या अपना प्रश्न टाइप करें या बनाएं। हर चरण, एक चित्र, और उत्तर तब तक छिपा रहता है जब तक आप न पूछें।
एक स्वतंत्र वृत्तान्त प्रत्येक पाठ पर आगे कहता है, जो आपने पूरा किया है, आपके हल किये गये समस्याओं का रिकॉर्ड, और एक शिक्षक जिसके बारे में आप पूछ सकते हैं ।
साइन अप करें लॉगिनयहाँ उपयोग में संकेत
पूरी परिभाषा, चित्र, और इसमें हर अक्षर का अर्थ क्या है?
कैसे: σ-algebras and measures
- First find an antiderivative F, then evaluate F(b) − F(a).
- The integral of a constant c is c·x.
- Fundamental theorem of calculus: plug in the limits.
- Simplify.
लोग जो सवाल पूछते हैं
What is wrong with the Riemann integral?
It fails on functions that oscillate too much, and it does not interact well with limits: the limit of integrable functions need not be integrable. Lebesgue's integral fixes both.