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# 数学代写|随机分析代写Stochastic Analysis in Finance代考|MA3210 Commutative Hypergroups

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## 数学代写|随机分析代写Stochastic Analysis in Finance代考|Deﬁﬁnition and ﬁﬁrst examples

In this introductory section we provide the standard notions of hypergroup theory and a few examples which show how certain classes of hypergroups originate from groups. Some conventional agreement seems to be in order. For a locally compact space $K$ we apply the abbreviations $C(K), C^b(K), C^0(K)$ and $C^c(K)$ for the spaces of continuous functions, bounded continuous functions, continuous functions vanishing at infinity, and continuous functions with compact support on $K$ respectively. With the symbol $B(K)$ for the space of Borel measurable functions on $K$ we obtain the obvious sequence of inclusions
$$C^c(K) \subset C^0(K) \subset C^b(K) \subset C(K) \subset B(K) .$$
Similarly we consider the inclusions
$$M_c(K) \subset M^b(K) \subset M(K)$$
and
$$M^1(K) \subset M^b(K),$$
where $M(K), M^b(K), M_c(K)$ and $M^1(K)$ denote the sets of all (Radon) measures, of bounded measures, measures with compact support and of probability measures on $K$ respectively.

## 数学代写|随机分析代写Stochastic Analysis in Finance代考|Some harmonic analysis

The analysis of hypergroups $K$ depends as in the group case on the existence of (left) Haar measures in $M_{+}(K)$ which by definition are nonvanishing and $T^x$-(left) invariant for each $x \in K$. Haar measures exist on hypergroups as long as they are discrete or compact or commutative. They are unique within a positive constant and have full support. So, for a given hypergroup $K$ its Haar measure $\omega_K$ will always be fixed.

If $K$ is a compact hypergroup, $\omega_K$ is necessarily bounded, hence it can be normalized, and it is an idempotent measure (in the sense of convolution) in $M^1(K)$. Moreover, any hypergroup admitting a bounded Haar measure $\omega_K$ must be compact.
FROM NOW ON WE ASSUME $(K, *)$ TO BE COMMUTATIVE.
1.2.1. Definition
For $\mu \in M^b(K)$ and $g \in B(K)$ the convolution of $\mu$ with $g$ is given by
$$\mu * g(x):=\int_K g\left(y^{-} * x\right) \mu(\mathrm{d} y)$$
for all $x \in K$.
If $f, g \in B(K)$ with at least one of the functions $f, g$ having $\sigma$-finite support, then the convolution of $f$ with $g$ is defined by
$$f * g(x):=\int_K f(x * y) g\left(y^{-}\right) \omega_K(\mathrm{~d} y)$$
whenever $x \in K$.
The convolution of functions gives rise to the introduction of the spaces $L^p(K):=L^p\left(K, \omega_K\right)$ for $1 \leq p \leq \infty$.

## 数学代写|随机分析代写金融中的随机分析代考|定义和fifirst例子

$$C^c(K) \subset C^0(K) \subset C^b(K) \subset C(K) \subset B(K) .$$

$$M_c(K) \subset M^b(K) \subset M(K)$$

$$M^1(K) \subset M^b(K),$$
，其中$M(K), M^b(K), M_c(K)$和$M^1(K)$分别表示$K$上的所有(Radon)度量集、有界度量集、紧支撑度量集和概率度量集

## 数学代写|随机分析代写金融中的随机分析代考|一些调和分析

1.2.1。对于$\mu \in M^b(K)$和$g \in B(K)$, $\mu$与$g$的卷积由
$$\mu * g(x):=\int_K g\left(y^{-} * x\right) \mu(\mathrm{d} y)$$

$$f * g(x):=\int_K f(x * y) g\left(y^{-}\right) \omega_K(\mathrm{~d} y)$$

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