### Journals Information

**
Mathematics and Statistics Vol. 11(3), pp. 446 - 453 DOI: 10.13189/ms.2023.110302 Reprint (PDF) (811Kb) **

## The Form of σ-Algebra on Probability Hilbert Space

**Bernadhita Herindri Samodera Utami ^{1}^{,2}, Mustofa Usman ^{3}, Warsono ^{3}, Fitriani ^{3}^{,*}**

^{1}Doctoral Program of Mathematics and Natural Science, Faculty of Mathematics and Natural Science, Universitas Lampung, Bandar Lampung, 35145, Lampung, Indonesia

^{2}Department of Information System, Institut Bakti Nusantara, Pringsewu, 35373, Lampung, Indonesia

^{3}Department of Mathematics, Universitas Lampung, Bandar Lampung, 35145, Lampung, Indonesia

**ABSTRACT**

Measure theory is used as the basis for probability theory. One of the most useful measure theories for statistics and probability theory is the concept of distance. The concept of distance introduced in the inner product space is closely related to the order relation in each sequence of elements. In statistics, random variables can be seen as a sequence that can be an object to study, including the partial ordering relation, expectation value, convergence, also infimum and supremum. This study aims to obtain the properties of a partial ordering relation which is useful for forming probability Hilbert spaces, more specifically the σ-algebra. If in ordinary sets, σ-algebra uses the concept of intersection and combination of sets, in probability Hilbert space, σ-algebra uses the concept of partial relation ordering, lattice, and indicator lattice. This research is quantitative research with a method of proof to generalize the concept of the order of elements. The novelty of this research is to find the associative properties of lattice in Hilbert probability space as described in Corollary 1. Furthermore, based on the definition of absolute value in Hilbert probability space, we derive the properties of addition and subtraction of absolute values and find their relationship with the lattice stated in Proposition 1. In the Hilbert probability space, the convergence property of random variables also applies which results in the lattice convergence stated in Proposition 2. Finally, it can be shown that the set of indicators in the Hilbert probability space form the algebra σ which is stated in Proposition 3. This study also gave use of the dataset shares of 42 energy companies in Indonesia in 2022. The results of plotting the data using the probability density function of the Normal distribution, Log-Normal distribution, and Cauchy distribution.

**KEYWORDS**

Hilbert Space, Lattice, Partial Ordered Relation, σ-Algebra

**Cite This Paper in IEEE or APA Citation Styles**

(a). IEEE Format:

[1] Bernadhita Herindri Samodera Utami , Mustofa Usman , Warsono , Fitriani , "The Form of σ-Algebra on Probability Hilbert Space," Mathematics and Statistics, Vol. 11, No. 3, pp. 446 - 453, 2023. DOI: 10.13189/ms.2023.110302.

(b). APA Format:

Bernadhita Herindri Samodera Utami , Mustofa Usman , Warsono , Fitriani (2023). The Form of σ-Algebra on Probability Hilbert Space. Mathematics and Statistics, 11(3), 446 - 453. DOI: 10.13189/ms.2023.110302.