Techniques in Mathematics

Transformation to more comfortable form  

1. Laplace Transformation 

2. Talyor theorem 

3. Fourier transform 

4. MVT if continuous and differentiable 

5. Characteristic function or Simple function 

6. Fundamental Theorem of Calculus 

Analysis 

1. Bound

If $|a_{n}| < \epsilon$ and want to calculate $\sum_{n=1}^{\infty}a_{n}$, bound $|a_{n}| < \frac{\epsilon}{2^n}$ for $\sum_{n=1}^{\infty}a_{n} < \epsilon$

2. Show existence 

1) Construction 

- Collecting sets having common property that we want to show, then observe what common properties they are sharing. 

- For measure construction, always starts from outer measure. 

2) Approximation 

- Arzela Ascoli 

- $f \in \beta$, $f \in L_{1}$, then there exists a continuous $g$ with compact support such that $\int |f-g| < \epsilon$ 

- Simple 

- Egorov 

- For Lebesgue measure on $\mathbb{R}$, for any interval, there exists open and compact sets to approximate such an interval 

- Radon derivative 

- Stone Weierstrass Theorem 

3. Inequality 

- Jensen ==> Linear 한 관계 with 합치면 1되는 일 때 

- Chebyshev Inequality 

- Convex (Concave) 

- Houlder 

- Minkowski 

- Young 

4. Transformation 

- 곱했다 나누기 

- 더했다 빼기 

- FTC 

- Fubini : 조금 더 편한 measure 로 컨트롤하기 

- Change of variable