Кіріспе
Нақты мәнді функциялардың жіктелуі
Decomposition of real valued functions
In mathematics, the positive part of a real or extended real valued function is defined by the formula
Intuitively, the graph of is obtained by taking the graph of , chopping off the part under the x axis, and letting take the value zero there. Similarly, the negative part of f is defined as
Note that both f^(+) and f^(−) are non negative functions. A peculiarity of terminology is that the 'negative part' is neither negative nor a part (like the imaginary part of a complex number is neither imaginary nor a part). The function f can be expressed in terms of f^(+) and f^(−) as
Also note that
Using these two equations one may express the positive and negative parts as
Another representation, using the Iverson bracket is
One may define the positive and negative part of any function with values in a linearly ordered group. The unit ramp function is the positive part of the identity function.
In mathematics, the positive part of a real or extended real valued function is defined by the formula
Intuitively, the graph of is obtained by taking the graph of , chopping off the part under the x axis, and letting take the value zero there. Similarly, the negative part of f is defined as
Note that both f^(+) and f^(−) are non negative functions. A peculiarity of terminology is that the 'negative part' is neither negative nor a part (like the imaginary part of a complex number is neither imaginary nor a part). The function f can be expressed in terms of f^(+) and f^(−) as
Also note that
Using these two equations one may express the positive and negative parts as
Another representation, using the Iverson bracket is
One may define the positive and negative part of any function with values in a linearly ordered group. The unit ramp function is the positive part of the identity function.
Математикада, нақты немесе кеңейтілген нақты мәнді функцияның оң бөлігі мына формуламен анықталады:
Decomposition of real valued functions
In mathematics, the positive part of a real or extended real valued function is defined by the formula
Intuitively, the graph of is obtained by taking the graph of , chopping off the part under the x axis, and letting take the value zero there. Similarly, the negative part of f is defined as
Note that both f^(+) and f^(−) are non negative functions. A peculiarity of terminology is that the 'negative part' is neither negative nor a part (like the imaginary part of a complex number is neither imaginary nor a part). The function f can be expressed in terms of f^(+) and f^(−) as
Also note that
Using these two equations one may express the positive and negative parts as
Another representation, using the Iverson bracket is
One may define the positive and negative part of any function with values in a linearly ordered group. The unit ramp function is the positive part of the identity function.
In mathematics, the positive part of a real or extended real valued function is defined by the formula
Intuitively, the graph of is obtained by taking the graph of , chopping off the part under the x axis, and letting take the value zero there. Similarly, the negative part of f is defined as
Note that both f^(+) and f^(−) are non negative functions. A peculiarity of terminology is that the 'negative part' is neither negative nor a part (like the imaginary part of a complex number is neither imaginary nor a part). The function f can be expressed in terms of f^(+) and f^(−) as
Also note that
Using these two equations one may express the positive and negative parts as
Another representation, using the Iverson bracket is
One may define the positive and negative part of any function with values in a linearly ordered group. The unit ramp function is the positive part of the identity function.
Интуитивті түрде, графигін алып, x осьінің астындағы бөлігін кесіп тастап, сол жерде мәнін нөлге теңеу арқылы графигі алынады. Сол сияқты, f функциясының теріс бөлігі былай анықталады:
Decomposition of real valued functions
In mathematics, the positive part of a real or extended real valued function is defined by the formula
Intuitively, the graph of is obtained by taking the graph of , chopping off the part under the x axis, and letting take the value zero there. Similarly, the negative part of f is defined as
Note that both f^(+) and f^(−) are non negative functions. A peculiarity of terminology is that the 'negative part' is neither negative nor a part (like the imaginary part of a complex number is neither imaginary nor a part). The function f can be expressed in terms of f^(+) and f^(−) as
Also note that
Using these two equations one may express the positive and negative parts as
Another representation, using the Iverson bracket is
One may define the positive and negative part of any function with values in a linearly ordered group. The unit ramp function is the positive part of the identity function.
In mathematics, the positive part of a real or extended real valued function is defined by the formula
Intuitively, the graph of is obtained by taking the graph of , chopping off the part under the x axis, and letting take the value zero there. Similarly, the negative part of f is defined as
Note that both f^(+) and f^(−) are non negative functions. A peculiarity of terminology is that the 'negative part' is neither negative nor a part (like the imaginary part of a complex number is neither imaginary nor a part). The function f can be expressed in terms of f^(+) and f^(−) as
Also note that
Using these two equations one may express the positive and negative parts as
Another representation, using the Iverson bracket is
One may define the positive and negative part of any function with values in a linearly ordered group. The unit ramp function is the positive part of the identity function.
Ескеріңіз, және теріс емес функциялар болып табылады. Терминологияның ерекшелігі – «теріс бөлік» теріс те емес, бөлік те емес (кешен санның жолғасы жолғас та емес, бөлік те емес сияқты). f функциясын және арқылы келесідей көрсетуге болады:
Decomposition of real valued functions
In mathematics, the positive part of a real or extended real valued function is defined by the formula
Intuitively, the graph of is obtained by taking the graph of , chopping off the part under the x axis, and letting take the value zero there. Similarly, the negative part of f is defined as
Note that both f^(+) and f^(−) are non negative functions. A peculiarity of terminology is that the 'negative part' is neither negative nor a part (like the imaginary part of a complex number is neither imaginary nor a part). The function f can be expressed in terms of f^(+) and f^(−) as
Also note that
Using these two equations one may express the positive and negative parts as
Another representation, using the Iverson bracket is
One may define the positive and negative part of any function with values in a linearly ordered group. The unit ramp function is the positive part of the identity function.
In mathematics, the positive part of a real or extended real valued function is defined by the formula
Intuitively, the graph of is obtained by taking the graph of , chopping off the part under the x axis, and letting take the value zero there. Similarly, the negative part of f is defined as
Note that both f^(+) and f^(−) are non negative functions. A peculiarity of terminology is that the 'negative part' is neither negative nor a part (like the imaginary part of a complex number is neither imaginary nor a part). The function f can be expressed in terms of f^(+) and f^(−) as
Also note that
Using these two equations one may express the positive and negative parts as
Another representation, using the Iverson bracket is
One may define the positive and negative part of any function with values in a linearly ordered group. The unit ramp function is the positive part of the identity function.
Сонымен қатар, мына екі теңдеуді пайдаланып, оң және теріс бөліктерді былай көрсетуге болады:
Decomposition of real valued functions
In mathematics, the positive part of a real or extended real valued function is defined by the formula
Intuitively, the graph of is obtained by taking the graph of , chopping off the part under the x axis, and letting take the value zero there. Similarly, the negative part of f is defined as
Note that both f^(+) and f^(−) are non negative functions. A peculiarity of terminology is that the 'negative part' is neither negative nor a part (like the imaginary part of a complex number is neither imaginary nor a part). The function f can be expressed in terms of f^(+) and f^(−) as
Also note that
Using these two equations one may express the positive and negative parts as
Another representation, using the Iverson bracket is
One may define the positive and negative part of any function with values in a linearly ordered group. The unit ramp function is the positive part of the identity function.
In mathematics, the positive part of a real or extended real valued function is defined by the formula
Intuitively, the graph of is obtained by taking the graph of , chopping off the part under the x axis, and letting take the value zero there. Similarly, the negative part of f is defined as
Note that both f^(+) and f^(−) are non negative functions. A peculiarity of terminology is that the 'negative part' is neither negative nor a part (like the imaginary part of a complex number is neither imaginary nor a part). The function f can be expressed in terms of f^(+) and f^(−) as
Also note that
Using these two equations one may express the positive and negative parts as
Another representation, using the Iverson bracket is
One may define the positive and negative part of any function with values in a linearly ordered group. The unit ramp function is the positive part of the identity function.
Тағы бір жазу тәсілі, Iverson жақшасын қолдана отырып:
Decomposition of real valued functions
In mathematics, the positive part of a real or extended real valued function is defined by the formula
Intuitively, the graph of is obtained by taking the graph of , chopping off the part under the x axis, and letting take the value zero there. Similarly, the negative part of f is defined as
Note that both f^(+) and f^(−) are non negative functions. A peculiarity of terminology is that the 'negative part' is neither negative nor a part (like the imaginary part of a complex number is neither imaginary nor a part). The function f can be expressed in terms of f^(+) and f^(−) as
Also note that
Using these two equations one may express the positive and negative parts as
Another representation, using the Iverson bracket is
One may define the positive and negative part of any function with values in a linearly ordered group. The unit ramp function is the positive part of the identity function.
In mathematics, the positive part of a real or extended real valued function is defined by the formula
Intuitively, the graph of is obtained by taking the graph of , chopping off the part under the x axis, and letting take the value zero there. Similarly, the negative part of f is defined as
Note that both f^(+) and f^(−) are non negative functions. A peculiarity of terminology is that the 'negative part' is neither negative nor a part (like the imaginary part of a complex number is neither imaginary nor a part). The function f can be expressed in terms of f^(+) and f^(−) as
Also note that
Using these two equations one may express the positive and negative parts as
Another representation, using the Iverson bracket is
One may define the positive and negative part of any function with values in a linearly ordered group. The unit ramp function is the positive part of the identity function.
Кез келген мәндері сызықты реттелген топта болатын функцияның оң және теріс бөліктерін анықтауға болады. Бірлік рампа функциясы – сәйкестік функциясының оң бөлігі.
Decomposition of real valued functions
In mathematics, the positive part of a real or extended real valued function is defined by the formula
Intuitively, the graph of is obtained by taking the graph of , chopping off the part under the x axis, and letting take the value zero there. Similarly, the negative part of f is defined as
Note that both f^(+) and f^(−) are non negative functions. A peculiarity of terminology is that the 'negative part' is neither negative nor a part (like the imaginary part of a complex number is neither imaginary nor a part). The function f can be expressed in terms of f^(+) and f^(−) as
Also note that
Using these two equations one may express the positive and negative parts as
Another representation, using the Iverson bracket is
One may define the positive and negative part of any function with values in a linearly ordered group. The unit ramp function is the positive part of the identity function.
In mathematics, the positive part of a real or extended real valued function is defined by the formula
Intuitively, the graph of is obtained by taking the graph of , chopping off the part under the x axis, and letting take the value zero there. Similarly, the negative part of f is defined as
Note that both f^(+) and f^(−) are non negative functions. A peculiarity of terminology is that the 'negative part' is neither negative nor a part (like the imaginary part of a complex number is neither imaginary nor a part). The function f can be expressed in terms of f^(+) and f^(−) as
Also note that
Using these two equations one may express the positive and negative parts as
Another representation, using the Iverson bracket is
One may define the positive and negative part of any function with values in a linearly ordered group. The unit ramp function is the positive part of the identity function.
Өлшемдік-теориялық қасиеттері
Өлшеуге болатын кеңістік (X, Σ) берілген жағдайда, кеңейтілген нақты мәнді f функциясы өлшеуге болады, егер және тек қана оның оң және теріс бөліктері өлшеуге болатын болса. Сондықтан, егер f функциясы өлшенетін болса, оның абсолюттік шамасы да өлшенетін болады, себебі ол екі өлшенетін функцияның қосындысы. Алайда, керісіншесі міндетті түрде орындалмайды: мысалы, егер f функциясын V Vitali жиыны ретінде алсақ, f функциясы өлшенбейтіні анық, бірақ оның абсолюттік шамасы тұрақты функция болып табылады. Функцияның оң және теріс бөліктері нақты мәнді функция үшін Лебег интегралын анықтау үшін қолданылады. Осы функцияны ыдыратуға ұқсас, белгілі бір шамадағы өлшемді де оң және теріс бөліктерге жіктеуге болады – Хан ыдырау теоремасын қараңыз.
Given a measurable space (X, Σ), an extended real valued function f is measurable if and only if its positive and negative parts are. Therefore, if such a function f is measurable, so is its absolute value , being the sum of two measurable functions. The converse, though, does not necessarily hold: for example, taking f as
where V is a Vitali set, it is clear that f is not measurable, but its absolute value is, being a constant function. The positive part and negative part of a function are used to define the Lebesgue integral for a real valued function. Analogously to this decomposition of a function, one may decompose a signed measure into positive and negative parts — see the Hahn decomposition theorem.
where V is a Vitali set, it is clear that f is not measurable, but its absolute value is, being a constant function. The positive part and negative part of a function are used to define the Lebesgue integral for a real valued function. Analogously to this decomposition of a function, one may decompose a signed measure into positive and negative parts — see the Hahn decomposition theorem.